Back to you want them isolated from the bike's main lights. If properly isolated and the bike lights function properly without trailer, connected won't matter. The isolator takes the signal and gives it the power from your battery.
If the lights don't function properly without the trailer you need to check light connections, wiring, relays. . . on the bike.
On the trailer side if the bike is fine but trailer is erratic, I go back to solder and heat shrink, run check of wire for chafing and push thru wire loom , run a ground wire to your lights and you will be much happier with the reliability. After many years with boat, jet ski, farm and utility trailers I can tell you grounds are the source of many trailer woes.
If they put all led's in the back and stuck a load equalizer in and you are grabbing behind that it could be your issue, you want the wires to come from under your back seat.