I think you can:
1) Buy an external 24V charger and just run an extension cord from your EU1000 to it to charge your batteries on the occasions you need to (probably least expensive option since you would only be buying the charger. The example charger I posted a while ago is $99 and pushes 10A. There are a lot of other options out there.)
2) Replace your inverter with an inverter/charger that has an automatic transfer switch. Then wire that up so that you have the appropriate generator input jack on the side of the cabin. In this scenario you still fire up your genny, run an extension cord and plug something in when you need to charge the batteries. The inverter/chargers w/ATS aren't cheap. Here's an example:
Find many great new & used options and get the best deals for Samlex EVO-2224E 230V Pure Sine Inverter Charger With Battery Charger at the best online prices at eBay! Free shipping for many products!
www.ebay.com
There are a ton of options on that front as well.