Well, first thing is that 7Ah equals 7000mAh, and a 3-cell (3S) Lithium battery is rated at 11.7 volts, fully charged it will be about 12.4 volts. Since you probably have no idea what the internal resistance is of your vest, can you tell us approximately HOW LONG your fully-charged lead-acid battery will last (continuous usage)? That will let us guess at the amperage your setup will consume energy, so maybe we could make an intelligent suggestion.
However, Lithium polymer batteries are NOT designed to be drained of all their power, as it will readily damage the battery. To you, that means you would need to install an electrical cutoff circuit that would shutdown when the battery was drained to (3.0 X 3 = ) 9.0 volts, ABSOLUTE MINIMUM! I'll bet you won't be very happy when it shuts off at 10PM! So, you'd better plan to carry a SPARE LiPo battery also, in your backpack.
Also, there is a limit to the amperage drainage that a LiPo battery can maintain! I'm sure you've heard of the fires that can happen when they get shorted out.
Anyway, we'll need the above info from you, if you're still interested.