Your description of how/why you think the start battery isn’t charging is a bit suspect. Because if it starts it starts. And has not sure how battery charge is related to fuel. IE it isn’t.
But is that why you drive it “a few miles every 10 days”? To charge the battery? Also an odd routine. And a few miles won’t do much ifnthe battery is already low.
Why not just throw a battery maintainer on it and not worry about it? That’s the cheapest easiest solution and requires no time nor effort for your periodic short drives you’re doing now.