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That depends on your power consumption. How many hours of on time, how many lights, and how much time will your engine be running so the alternator is putting out, but my first thought would be just try it and see. It won't hurt anything and I suspect you will be fine.
Well, your LED strips should have a watts/foot rating, so you can get a pretty good idea of what your power draw will be. How many hours/day and for how many days between driving/recharging do you think you'll be using them?
Probably fine, but if you like numbers, that's the kind of thinking you should start with.
Have a lashup similar to that, a deep-cycle battery on a trailer to run a fuel pump (WAY more draw than what you're doing). Seems to work fine, as long as there's enough run-time between tanks of fuel to charge it back up. There is some voltage drop, but not much, the circuit's still well over 12v at the battery - heavy gauge charge wire helps the drop. Big thing is to remember that you have a hot wire in the trailer plug all the time - I have thought about rigging mine on a relay or diode or something of the sort to kill it when you unplug from the pickup, but haven't done it yet.....