External level too high and exasperated by the single course of bricks with thick cement on top , no sign of dpc, or weep holes which are especially important with an inward opening door.?
I fitted a fan in third floor bathroom , used a length of string thru the hole to pull up outer grill and fixed with silicon thru the hole and cable ties to hold in position.
Pair of kitchen wall brackets with one reversed would work . Or use the plate you have and only fix at top and bottom holes, back off screw half a turn which will allow lateral movement .
Currently have 3m timber from a dozen pallets or more , very common where I live .
Last one I picked up used 3x2 and 4x2 timber and a couple of 8x2 timbers.
I just put a small screw into trim, texture of trim usually prevents tape sticking.Or you could stick a couple of layers of foil tape on trim which leaves smooth surface for tape to stick too.