I use one of the HF chargers too. I have measured it MANY times, and I continue to measure it often (I was always taught "Trust, but verify" and my trust in cheap tools is thin). It has always put out a nice float level of 13.2V. Though I smelled something funny in the garage this morning leaving for work, so I'll have to check up on that battery.
That being said, if I were able to plug my trailer in while in storage, I would just depend on the built-in charger to maintain the battery, as I have verified that it is in fact a multi-stage charger, and have also verified its voltage output at all stages.
If your trailer is older and does not have a multi-stage charger, I would not leave the battery plugged into it.
2014 Jay Flight 28 BHBE
2015 RAM 2500 6.4L HEMI, Tradesman 4x4, 3.73
Blue Ox SwayPro (BXW 1503)
Upgraded from an REI internal frame backpack and a Eureka 1/2 dome tent!