How to lay LVT to top stair tread, transition strip, stair nosing?

[YorkieSelfBuild]

Hi, I'm laying Karndean Rigid LVT upstairs and wondering how to handle the top tread of stairs. I'll be painting the stairs, instead of carpet.

 

I've found a stair nosing that goes over the stair tread but the section that the LVT slides into looks about 6mm deep, not enough for a 5mm expansion gap and cover enough of the LVT. https://lilleytileandstone.co.uk/genesis-tvb-aluminium-vinyl-bullnose-profile-see-options.html

 

Would a standard transition strip work, those used for doorways? Though I might need a stair nosing for the top tread to make it look nice.

 

I'll post photos when I get home.