I noticed you used the comment about using the charging cables provided with your Honda EU30001s generator to re-charge your battery..
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You cannot expect much charging for your battery by using this 12V DC CHARGING OUTPUT on the front panel on your generator. I know the manuals calls this a DC CHARGING port but in reality this will take forever and probably two more days to re-charge a battery. The output here is most likely unregulated and will be only around 12VDC and only just a few AMPS output.
Yes this will charge your battery but it will take along time. I could not imagine running my generator for over 24 hours to charge up a deep cycle battery. My 2KW honda has a similar 12VDC charge port but it will only run around 7 hours on one tank of fuel haha...
What you want to do is plug your 30AMP SHore Power Cable directly into your Honda 3000 generator and allow the trailer on-board smart mode CONVERTER to re-charge your trailer battery just like you do when plugged into shore power.
Your trailer being a 2010 model should have the SMART-MODE TECHNOLOGY CONVERTER/CHARGER installed and if so so this will re-charge your battery back up to 90% charge state in only three hours run time of the generator. What makes this work for you is the smart mode converter uses 14.4VDC to start charging your battery and in about two hours will shift down to 13.6VDC for another hour which will get your battery to at least a 90% charge state. The battery will require around 12 hours of charge to attain a 100% state of charge.
The nature of the Deep Cycle Batteries is you can start using them at the 90% charge state and expect their rated specified amp hour capacity without doing harm to the battery. When I am camping off the power grid I can do around ten of the these 50% to 90% charge states cycles while camping each day but will need to do a full 100% charge routine in ten days or so to keep from doing internal damage to the battery. The deep cycle batteries are designed to do this for you during the lifetime of the battery.
The only good use I have ever found to use the Battery Charge plug for is to run some DC lights around the generator when it gets dark to see how to re-fuel the generator haha... The 12VDC charge output does not do anything for me battery charging wise.
Just passing this along FYI...
Roy Ken