If I were in your position, I would absolutely repot. 6" deep is definitely not too deep.
I have a pot that's close to 14" deep/wide with some relatively small flytraps in it and they thrive. For now 6" is plenty. Maybe repot to 8" or 10" next
Spring depending on how big the flytrap is getting.
If it gets enough light (e.g. 6 hours or more sun per day) it will recover fine from a repotting; less sun than that and it may take longer to recover. It's still early enough in the season that it shouldn't be much of a problem.
Its growth will completely stop for a couple weeks after a repotting (due to shock), so keep that in mind. The pot size you mentioned would be great as long as it's plastic/glass/styrofoam. Just clean it out really well so any previous media is washed out completely, or as good as you can possibly get it.
The bigger pot will allow it to more easily divide this year so by the end of the year in that bigger pot (with more room for its roots to expand) you may end up with two or even three flytraps.
Be gentle with your flytrap's roots - messing with their roots is the reason they go into shock after repotting so the less you stress the roots, the faster the plant will recover. Try not to break any roots, as this will add to the shock, but don't stress if you do. I break roots on my flytraps all the time repotting and none have died - they just refuse to grow for a little longer.
As far as media, this article will explain: http://www.flytrapcare.com/media-soil-f ... traps.html
(let us know if you have other questions)
I'll have to defer to someone who's in the UK for any specific brand recommendations, but in general as long as the peat/perlite/or silica sand is fertilizer-free with no "added nutrients" or anything like that, it should be fine.