Would think Stixall (from competitor) would stick it. If it's hollow then I would squirt some low expansion foam in there to give it some support before gluing. https://www.everbuild.co.uk/product/stixall/
Or clad over with upvc 90 degree angle, trimmed to correct size As Kiab says, fill hollows (also between old and new trim) with silicon to make solid once set as peeps do tend to step on this part of frame Done neatly (as you would) ok, a different profile to frame but would people notice ?
This is from the outside looking in. I have considered sticks like but as already said it gets trodden on and is hollow behind.