With future kits, you might find that you don't need to swab your tonsils anyway.
I've been doing lateral flow test twice per week since April. The ones I was initially given by my local pharmacy all had long swab sticks in, with the "four on each tonsil and then up your nose" instructions. However more recently I've only been getting kits with short swab sticks, with "five times round up each nostril" (with no throat swabs) for the instructions.
Given that the kits with long swab sticks always mentioned the 'nose only technique' as a back up plan for people who had problems testing themselves (or their children) using throat swabs, and that kits are now being issued with only nasal testing shown in the instructions anyway, it would seem safe enough (in my opinion) to assume that it's a valid way of testing.
Here are the instructions that I've got with a 'short swab' kit:
https://postimg.cc/XXhydmTy
You'll see that it's specifically stated that only nasal swabs should be taken.
The instructions for 'washing' the swab in the fluid require a longer time to swirl it around though
https://i.postimg.cc/dt81B7rx/Scan-Image002.jpg
and four drops of fluid, rather than two, are used for the test.
So using your 'long swab' kit as if it were as 'short swab' one might be all that you need to do. (In future ask you pharmacy if they've got Flowflex testing kits, as they're the ones with the short swabs).