First thing first, measure the battery voltage right before you turn the lights on and again once they start blinking.
Most small batteries are just that, small and at 30w of draw that's almost 3 amps per hour. IF your battery is a deep cycle and not just a bog standard lawnmower battery then that's about half the capacity in the first hour. If it's just a normal battery they're designed to release a whole lotta power all at once then get recharged and can't really maintain a load for very long.
By the sounds of it the battery is discharging to the cut off voltage setting of the PWM (I'm guessing the lights are on the LOAD port of the SCC?) and the batteries cut off. Then they recover right above the cut-off and come on, which drops them right below the cut-off and rinse, repeat. To calculate a battery you'll need to do some math.
Napkin math says 30w @ 12v - efficiency = 3A/hr. $Running hours * 3amps / hr = capacity required. Lead Acids are good for 50% DoD, so 3A/hr * $RunningHours * 2 = Battery Capacity in Ah rating, or simplified is $RunningHours * 6 = Ah.
Example: You want the lights to stay on all night long which in winter is 14 hours. The lights require 30 watts and your battery is a WallyWorld Deep Cycle because it was cheap. 30w * 14hr = 420Watt-Hours to keep the lights on all night. Since your system runs at 12v that's 420Wh / 12v = 35Ah. Since you're using led acid because it's cheap and available, you only get 50% out of the rating on the sticker in Ah, so 35Ah *2 = 70Ah, You'd need a 70Ah battery to get through 1 night and have to fully charge it the next day. If you wanted the lights on and the weather was krappy you'd need ANOTHER 70Ah of battery to make it 2 nights, or 140Ah total. Stupid math!
I tell people Watts are like Grams. It takes a LOT to do a little and nobody is gonna get ripped and totally swol on 5000 gram (or 11lb for us American types) weights.
Tl;Dr: I think your battery is too small and can't handle the load for long.