Short answer: you're probably fine to install it, especially if you only use one 5000+ connected to the STS.
Long answer: It all depends on what the electrician meant by "25 amps remaining".
The STS uses a 100 amp breaker. But, if you only have one 5000+ unit and only ever use one 5000+ unit, it will only draw a max of 47 amps. And the STS will only supply 30 amps. Still put in the 100-amp breaker of course, but just understand that one 5000+ won't ever be drawing 100 amps.
The thing is, electrical panels usually have extremely more breakers than they could ever possibly supply. As an example, my panel has a 150-amp main breaker; there's no way it can draw more than 150 amps because the breaker will trip. But in the panel, I've got a 40-amp, three 30-amp, about ten 20-amp, and about eight 15-amp breakers. So theoretically I should be able to connect appliances, lights, and tools that draw a total of about 450 amps -- yet that could never happen, because as soon as I draw 151 amps the main breaker will trip right away.
So the wiring in your house is way overkill. For example, I have a 15-amp circuit dedicated to the lights in the 2nd bathroom. Those lights draw about 0.01 amps total. Yet there's a whole 15-amp circuit dedicated to them. I will never, ever, ever reach that capacity. And that's more or less the case with pretty much every circuit; there's usually way more capacity than there ever will be usage. The exceptions would be huge power draws like the electric heat pump, water heater, and dryer; those are much more likely to use up most of their breaker capacity.
So what does the elec mean by "25 amps remaining"?
Scenario 1: something like "your main panel breaker is 200 amps, and all the breakers in your panel add up to 175 amps, so you have 25 amps remaining". In this case, you're almost certainly fine to add an STS, because you're extremely unlikely to be using all that capacity.
Scenario 2: something like "the electric heat pump and water heater are going to take up 175 amps, so you only have 25 amps remaining for everything in your house." I guess that's technically possible, but seems highly unlikely. A 2-ton heat pump generally uses about 37 amps, and electric water heaters generally use around 20 amps, so ... that math isn't mathing. Maybe you have a 30-amp electric oven/range, and other electric appliances, that bring you close to 175 amps of actual usage? Even then, that still seems unlikely, for reasons below.
So let me give you my scenario and you can see how it works:
I have a gas water heater, furnace, and stove. So those don't count. I have an electric dryer, and the 4-ton air conditioner is electric. On my STS, I put basically every circuit in the house other than the A/C and dryer on the STS. When the power goes out for an hour, I barely notice. We had a 5-hour outage here recently, and it did start getting warm (100 degree south Texas summer day, with no A/C). Wasn't bad, the ceiling fan helped a lot, but it was getting warm. Practically everything in the house worked just fine, the 5000+ and STS handled the TV, internet, refrigerator, lights, fans, computers, everything. And for all those appliances added together, it used an average of about 1000 watts total electricity. That's about 8 amps. We didn't do anything crazy, I didn't run my table saw and heat gun during that power outage for example, but we didn't huddle together around a candle either -- we just lived normally, modified only slightly by the grid being down. No laundry, for example.
All that was handled by the 5000+, easily. I didn't have solar at that time, so I did run my generator once to recharge the 5000+ when its battery charge got down to around 30%, since we had no idea how long the outage would last.
So here's the thing: the 5000+ supplied around 8 amps of power, sometimes more, sometimes less, mainly dependent on whether the refrigerator compressor was running or not. But those 8 amps don't count against you, because your house already supplies those 8 amps, whether you put them on an STS or not. The STS doesn't draw more amps to supply the house load than your house power already supplies. The STS only draws additional power to recharge the 5000+ battery. It can draw a max of 16.7 amps when recharging the 5000+.
That's why I say "you're probably fine to add an STS". At worst, it would draw 16.7 amps to recharge the 5000+, and the electrician already said you had 25 amps remaining (whatever was meant by that). But if you were to add a second 5000+ to your STS, then you *might* run into a potential issue because it could draw as much as 33.4 amps, which is above the 25-amp-remaining statement, depending on if that statement is to be read as "your house can only draw 25 more amps before the breaker trips". So the simple solution to that would be to use only one 5000+ until you get clarification from the electrician as to whether that meant "on a daily basis your appliances will always draw 175 amps, so don't add any more than 25 amps" or "the maximum possible if you used everything in your house all at once would draw 175 amps". Because we never do that, right? You don't run the dishwasher and the garbage disposal and the dryer 24 hours a day! And even then, even if you did run everything at once, the worst that will happen is the main breaker will trip. So shut some of those non-mandatory appliances off, and go reset the breaker -- it's a minor inconvenience.
Keep this in mind though: the 5000+ can only supply up to 30 amps on a continuous basis. And if it does supply that full 30 amps on a continuous basis, the battery would be completely drained in only about 40 minutes. So you wouldn't be putting the heat pump and the water heater and the electric oven and the electric dryer on the STS! Those are all huge draws that will drain it extremely quickly. The STS can handle 12 circuits. Put your emergency loads on the STS, things like the refrigerator and a CPAP machine and maybe your internet and computer (especially if you work from home) and some lights and outlets for any other emergency medical equipment, etc. If you have circuit space left over in the STS, add some convenience circuits, like more lights, the microwave and hair dryer. I doubt it gets very hot there, so the air conditioner probably wouldn't be a major factor, but a heater might. Any way to get heat that doesn't mean electricity would be preferable since electric heaters use enormous amounts of electricity. A propane heater would be efficient and wouldn't draw electricity, or if your house has natural gas service a gas heater would be a great backup heating source for long power outages.
Also, if you're expecting outages that may last longer than about 4 hours, you'll need a way to recharge the 5000+. Either solar, or through a generator. I bought a tri-fuel generator since my house has natural gas service, so I can run that thing all day every day for months on end and never have to refill it. Gasoline is okay but if you've got a multi-day outage gasoline might be hard to come by and cost quite a bit and there might be long lines, and at least down here all the blue rhino propane tanks are scarfed up almost immediately. If relying on solar, you'll have to cut loads at night down to the bare minimum so the 5000+ will last through the night, and hope that the day isn't cloudy. A generator is a very very good thing to have, just make sure you have fuel available for it.
Bummer! I hope someone figures it out, because it looks like a cool toy.
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