I think when the problem was happening with your indicator that was an earthing issue, sounds like that has been resolved with the earth strap.
It sounds like you already have enough load on the stop light circuit. If you have a resistor handy you can try putting it on the tail light.
Before you do that connect a voltmeter to one of the tail lights. For ease it can be the tail light feed in the trailer plug. With you foot on the brakes what is the reading?
Edit... the resistor fitted to the tail lights is only for testing purposes. It will generate too much heat if left fitted when the park lights were tuned on.