Very interesting.
It could be. I assume this solenoid is always on when the key is in 'run' or is it manually switched?
Normally, a low battery will draw a lot of current at first, but then as it charges less and less. High recharge current means lots of chemical activity in the battery and that produces heat. That internal heat makes it more difficult to recharge. The small wire on the fuse may restrict the recharging current somewhat, which is good but best would be to recharge that battery on a slow charger.
Anyway, its pretty easy to test the possibility by taking it off the circuit. Just watch where loose hot wires are!