If it's a 2-cell 8.4V battery, there's zillions of 2S chargers out there already; check thrift stores or yard sales for obsolete digicam chargers. But, I think it would be OK if you set the LM317 as constant voltage 8.4V (or a bit less if you believe in charging to 80% for maximum life), and added a series resistor to limit the current to a safe level, at the expense of slower charging. The battery should already have a protection module that will prevent dangerous overcharging, unless it's a cheap thing from China with an imaginary mAh rating.
Alternatively, there are dedicated lithium-ion charger chips. SY6912 is one that can do 1, 2 or 3 cells at 500 mA to 2A (configurable by jumpers and removing shunt resistors); I was favorably impressed, until one day I hooked the board up to the battery and smoke came out of the SY6912. Those boards are mostly what show up when you search ebay for 1s 2s 3s lithium charger; the designs look different than mine, so there may be variant on that chip, or a knockoff, or an improvement. I bought another to replace the dead board but haven't put it into use yet.
A more promising chip could be the MAX745 from a little company called Maxim. Those boards cost more (maybe that means they have genuine chips?), but have a dipswitch for setting a number of cells (1S to 4S), and trim pots for setting current and trimming voltage.
How small does your battery need to be? The 18650 form factor cells are common and can often be salvaged from recycled laptops and tool packs. They should be used with a module that provides balancing, over-charge, and over-discharge protection (those will be the boards that have rows of power MOSFETs and SMD resistors and chips with 4 to 6 legs. 3S (12.6V fully charged) is close enough to 12V for most purposes.
Alternatively, use a single lithium cell and a boost converter to power your circuit. That's common in many portable devices. Single cells are easy to charge from USB and don't need balancing. You may be able to find a spare battery for some old cellphone or salvage one from a dollar store item; those will nearly always include a protection board.

Hello all. I borrowed a car battery charger but am totally confused about what to buy for myself. I need to keep a 'resting' car battery topped up, and I would like to boost the stop-start version in my Tucson as it is only getting short runs lately. I see chargers advertised from about £15 to well over 10x that amount. Although I have access to mains for the caravan maybe that battery would benefit from charging maintenance out of the battery box too! What would readers recommend to me, please?

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Welcome to AAC!

You need something called a battery maintainer/restorer and they're going to cost more than £15.

I bought 4 of these for myself and one for my Son-in-Law:

I have 3 vehicles in storage with batteries that wouldn't hold a charge. One battery was 10 years old and is now holding a charge for weeks. Another was 11 years old and it has held a charge for more than 5 days so far. The third was 5 years old and now holds a charge for more than a month.

Previously I tried using a regular battery charger and running them often enough to keep the battery charged. The maintainers have restored life to batteries that are way out of warranty.

I had a battery for my riding lawnmower that I replaced a year ago. I put a maintainer on it and now it holds a charge for weeks; probably longer.

I put a new battery in my BMW and wish I had tried a maintainer on it before I did that. Anyway, I keep a maintainer on that battery because I haven't driven it much since the Pandemic. The last time I bought gas for it was over a year ago...

 

thank you so much for your suggestion

If the batteries are in really poor condition, the maintainer I referenced will say they're bad after trying to charge for an hour if they aren't holding a charge. If that happens, I keep trying the maintainer or use a normal charger and switch to the maintainer after they're charged up some.

I can hear a high pitched frequency from the maintainers while they're trying to remove sulfation. The state of charge indicator changes depending on the mode it's operating in; which changes automatically. The unit referenced will work for sealed or wet lead acid, 6V or 12V batteries, and has a special mode for winter.

It has not been able to restore any 12V SLA batteries taken from UPS's.