Wasn't expecting so many replies so fast. I use a minnkota trolling motor for docking and launching. Its only ever on for a minute or two per outing.
I think the 10 amp battery that I use with my 36lbs motor on my dinghy would work for a 2-3 sails on your boat. I get about 4-5 round trips out of mine. It takes about 3-4 minutes to get to my mooring. There's a huge difference in runtime and speed (stays up to speed better on trips 4 and 5) with the 10 amp versus a 7 amp. The lower capacity batteries I got maybe 3 round trips out of. Don't expect your trolling motor to get you out of heavy winds or waves.
I have a cheap Chinese solar controller dedicated to the dinghy battery on my boat. I have two batteries that get swapped. I found a plastic electrical box, while it doesnt look pretty, works well as a cheap, yet durable battery holder. Mine is stored below deck. I'm sure there are better looking battery boxes out there. For a small gel cell, I don't see a need for a lid. I'd use shock cord or cut up an old bungee to keep it in place. I'm sure you can find a plastic box of some sort that'll work well. Project boxes come to mind as do small equipment cases
There's quite a few bolts in this area that can be utilized to attach a battery box. You can even secure it with wood screws to the plywood if it not splitting like I think it is.
In your case, I'd probably put the battery box on the lower bolt, resting or nearly resting on the seat. It won't take up anymore room than the motor already is. In otherwords, your not losing usable space in your cockpit. My second thought was to utilize the bolts for the upper gudgeon (rudder mount). If it was my boat, I'd probably try to mount it in the cubby hole in the cockpit if there's access to route cabling in the hull
There's plenty of options and opinions to come. Let us know what you ultimately decide and we can provide you with advice on proper installation techniques