Oh wow. Some of the replies here are kind of nuts. You can put them in parallel, but there's no practical benefit. AGM can't "absorb" a voltage spike with any significant current. Unless a BMS has sizeable inductors (and I've never seen one that does), there would be nothing in a BMS to create a voltage spike upon failure or disconnection - I'll address this later. First off, look at the discharge curve for both chemistries:
I know it says Lithium-ion, but they mean LiFePo4. I pulled this off the interwebs to save time. But as you can obviously see, at no point do the two intersect, with AGM always lower, and 99.9% of the time significantly lower. So basically, the AGM battery will be a drag on the LFP battery. In other words, the LFP battery will always be trying to charge the AGM battery. On top of that, the internal resistance of the LFP battery is likely many times lower, meaning it can supply more current with less voltage drop - making the situation shown in the chart actually much worse under load. An LFP battery has much more usable energy capacity amp hr to amp hr as a result the shape of their discharge curve.
Here's a practical demonstration - take a standard lead acid car battery at 50% SOC and measure the voltage while cranking your car. Then take an LFP battery with half the amp hour rating (also at 50% SOC) and do it again - the voltage drop during cranking will practically always be a lot less than the lead acid battery, and your car will crank faster. Now if you wire the lead acid battery in parallel with the LFP battery, the LFP battery immediately starts charging the lead acid battery because of the voltage differential - about 12.8 vs 11.6 volts, repectively. So now you're draining the LFP battery instead of saving that energy to start your car. Once you hit the key, the LFP voltage will drop a bit, but not much (remember, very low internal resistance). If it does drop low enough to get to the lead acid battery's voltage level, then the lead acid battery will stop being a drag on the LFP battery, and breifly contribute, but since the lead acid battery's internal resistance is much higher than the LFP's internal resistance, its voltage will drop too, and remain a drag on the LFP battery. The internal resistance thing is why you can take a 5 lb lithium jump starter and crank your car like gange busters. But since the jump starter may only have a 5 to 10 amps hour capacity, it won't be able to crank your car as long as the 30-40 amp hour lead acid battery, though the lead acid battery will crank slower for longer - with voltage pretty soon dropping below the minimum necessary to fire the ignition at the same time - usually around 9-10 volts.
Putting them in series is akin to putting solar panels of different capacities in series - you'll only get the combined benefit of the lowest power panel.
I think you, the OP, have two practical options: use the LFP and re purpose the AGM for something else, or install a high current transfer switch and keep the AGM on a float charger and only use the AGM as emergency back up power. Which brings up charging, since the two chemistries like to be charged differently. You can float charge an AGM battery for a long time, and is in fact good for it - it minimizes sulfation, and therefore loss of capacity. However, floating an LFP can shorten its life dramatically; and LFP batteries live longest keeping the charge level at 80% or so.
To the guy doing this for "unattended" systems - you're not doing this correctly. Not only are you not doing anything to absorb "spikes" - which are likely not there, or if you're using some sort of esoteric BMS I'm not familiar with (some BMSs do have small inductors for cell balancing, but those would be too small to be significant - most BMSs use capacitive balancing (more expensive, but more efficient) or resistive (cheaper, typically more reliable, but waste energy, but I digress...)) Anyway, if you're using a weird BMS that does have inductor generated spikes upon failure (inductors are the only way to boost DC voltage, btw, and how boost converters work), be aware that such spikes are ALWAYS opposite polarity and are easily handled by putting a diode across the terminals to shunt such a spike - that's why most automotive relays have diodes built into them - a relay coil is basically an inductor after all, and when power is removed from the coil, the magnetic field in the core collapses and that energy (inductors store energy magnetically) is induced back into the coil in the opposite polarity. I would suggest a superfast diode for this purpose. For the low temp situation, you should be using a temperature controlled transfer switch or contactor to switch between the two battery types for all the reasons stated above.
