Look up SLA batteries. They are often used in electric razor scooters, emergency egress lights, backup power for commercial fire panels, etc. They are dirt cheap (often less than $5 each), they come in 6v, 12v, and sometimes 24v, and are often in the 4A/h range. That would get you half of a night on a 1A draw, so just get however many you need and wire them in series/parallel (depending on the voltage you get) until you have enough.
The 6v versions are also pretty small. Maybe the size of three decks of playing cards.
I have boxes of them at work. I use them for all sorts of things around the theater.
I've got ~30 automated water quality samplers scattered around the state. All have a 12V marine/RV battery powering them, and most have one of the 50W Renogy panels and PWM charge controller to keep them charged up. Most have a NAPA Legend group 31 battery, some have a pair of smaller batteries.
The Renogy stuff is cheap and prolific chinesium, you can find all over Amazon & eBay, but works well. Some of our older equipment has some $$$ Isco solar panels, but they provide no tangible benefit or advantage over the cheaper stuff, for what we need them for.
Renogy has changed their charge controllers over the years, most of ours are the older version of these. I just got 6 new controllers just like what come in these kits, last week to replace 4 that got flooded, and to have a couple spares.
http://https://www.amazon.com/gp/product/B00DCDVVO6/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
For even smaller scale stuff, I personally have a few of the 10W "Eco-Worthy" and "HQST" panels from Amazon paired with the Battery Tender charge controller. I've got 3, and I put together a pair for dad for Christmas last year, all keeping various project cars, lawn mowers, and farm equipment batteries charged up.
HQST 10W: https://www.amazon.com/gp/product/B072B5P91B/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
Eco-Worthy: https://www.amazon.com/gp/product/B00OZC3X1C/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
Battery Tender controller: https://www.ebay.com/itm/Battery-Tender-0211162-Solar-Panel-Charger-Controller/192497761466?hash=item2cd1c2bcba:g:IhcAAOSwv~lavyBz
