Just as another thought, you might consider a battery maintainer. Note that this is not the same as a "trickle charger", which is something to be avoided. You would need to disconnect the TM battery from the TM connections, and connect it to the maintainer's small cable. The maintainer can be mounted inside or outside the TM, as is most convenient for your situation. I don't think it is rain-proof, though.
The most important thing is that you do not leave the TM's built-in charger powered up and connected to the TM battery. Over a relatively long period of time, this does not work despite the advertising.
Among battery maintainers, I've had good luck with a Battery Tenda (aka Battery Tender). I bought the smallest one - 0.75 amp - and it is quite adequate provided you start with a fully charged battery. If you can't start with a fully charged battery, you might step up to the 1.5 amp version, but nothing higher is necessary. The 1.5 amp version will slowly bring your battery to full charge, and hold it there.
There are number of other battery tenders on the market that also produce good results. They have been discussed at some length here on the Forum. Use the search tool to find the discussions.
Bill
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