I cannot tell from your post just exactly which battery you intend to use. There are a lot of place to buy the small sealed lead-acid batteries in the general 12 volt 8 amp hour batteries cheaply, as low as $8-$10 each. Using one to full depletion is not going to kill it and even would it shorten the life substantially, you have so little into it that it wouldn't matter much.

The reason for a big heavy deep cycle battery is to gain the advantages of a the heavy plates inside to stand up to the thermal stress involved with heavey charge and discharge currents. A 36 lb thrust electric is not going to draw that king of current so you have half the problem that doesn't need solving. I solved the charging half by buying a 100 watt solar panel and keeping it connected while am out on the water. The battery I used was a small starting battery that goes in a Honda Civic. That battery is about 1/2 the weight of a standard deep cycle battery. I just bungee the panel onto the middle seat and I wind up with a "table" in the middle. Works superbly.I fish much larger water and a local river where I use the little troller for all it's worth and never had to hook a charger to it.all summer. I never ran out of battery on the water either. If you only ever use it for short periods the Panel with a regulator is the best way to charge any battery because it will not supply enough current to stress the battery unless you start using flashlight batteries.

I would seriously consider using a small car starting battery in this manner or a decent quality lawn tractor starting battery which is smaller and lighter yet, and will cost about what you probably paid for the Zeus. I have found that the much smaller load placed on batteries by fuel injected small cars tends to make the batteries much more likely to last 8-10 years than the much larger and much more heavily used batteries like e used to use in V8 engines. This extended life is because they start so much easier thus drawing much less out of the battery which means the charge current is much lower also. With the solar panel in the boat the system works the same way!