AGM batteries have a lower self discharge rate so might help, but make sure there is no current drain on the battery via a sensitive DC clamp on ampmeter. You shouldn't have to go AGM for a starter battery, but I use my retired home/shop 12V AGM 100AH battery for my Listeroid. I retire them to the Listeroid, with solar charger at 5 years. They may still have move capacity than needed for that application, since starting is via air motor. They die at about 10 years or so.
These clamp on DC/AC amp meters are cheap and work fairly well and can measure very small DC currents:
https://www.amazon.com/gp/product/B07P5QKQ5L/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&th=1You can also check the charging current easily with it.
An old lead acid battery may have a high self discharge rate, and it sounds like that's your problem, but I'd want to be sure before killing a replacement battery.
Taking out the battery and keeping it on a trickle charger is best for lead acid battery life. Solar panel and chargers work well as long as the panel is tilted to self clear of snow in a day or three, and the climate is suitable. I use one on my Kubota tractor. It's 7 year old battery loves it, since it's not used regularly. If you aren't getting 5+ years from a starting battery, you're killing it...most often by sulphation (protracted undercharge).
A modern switch mode low current charger off the mains as Mikenash suggests is quite high efficiency and is certainly no more of a load on the engine than the stock alternator charging circuitry. Fixing things is nice but it's a legitimate work around without penalty, especially if time and parts aren't available.
The cheapest $12 or less Chinesium on/off type solar charge controllers work well for your application; that's what I use for my Listeroid solar charger, with a roof mounted 40W PV panel.