On the face of it, there is no good answer. Ethonymy is literally folk etymology.
This does not give the curious etymologist a free pass to speculate wildly, but the semantics reflecting *kʷer- (as per your wikilink) vary so wildly that it looks like speculation to beginn with. This suits me well, because I want to elaborate an idea.
Pretext: Some labial plosives reconstruct labial velars in some languages (Greek, P-Celtic, some Italic tongues, ...). Greek kephale has not been related to Latin caput, which seems odd; wiktionary reconstructs *gʰebʰ-, and *kaput- respectively. Trying to ameliorate these, and always on a quest for German ge-, Lat. *co-, PIE *'ke-, *'kom, I did not get very far. I searched my Lexicon and at Greek bous, *gʷṓws the admittedly haphazard thought for ge- reappeared. Labio velars can occasionally come from *K + *w, so trying to separate the labial element from the velar I was grabbing for a straw. The nearest lexeme in my mind was still bous. I reason thus, a labiovelar can also come from *K + *b (or *b^h). That was not was I was looking for at all. Now to your question.
*kʷer- lists a Hittite reflex "to cut", and several others. It also lists synonyms *yeh₁- and *h₂er-. The latter reinforces my assumption that *kʷer- can have an internal derivation.
Following the pretext, look for *b(h) roots suitable for comparison; check
*bʰardʰéh₂, *bʰeh₁-, bʰeh₂-, *bʰreyH-, in that order. *bʰreyH- "to cut" rings a bell, of course.
Sure enough I can imagine backstories towards Briton for any given sense, but that's besides the point, I don't want to speculate. I only looked at *bʰreyH- because the link in the root index was dark-colored, visited before, but I didn't remember. That means it's a wild guess, except that *-r- clicked. Hence any further elaboration comes as an afterthought. Hittite is supposedly most archaic evidence, so I'm leaning more on the semantic side of it. Hittite also reflects *bʰreyH- but that doesn't change anything.
It only means that the root of Britain might go back a long way, and that a semantic interpretation is as good as irrecoverable without timely material evidence, because a loose semantic interpretation can always be given.
Point in case: a) To incorporate the Tattoo interpretation that wikipedia offers, suppose that roots meaning to write often come from "to cut", and tattoos are a form of writing. b) consider "Balto-Slavic: *kēr-, *ker-
Lithuanian: kẽras (“charm, magic”), kerė́ti (“to enchant, charm, bewitch”), kùrti (“to do, make, build”)
Slavic: *čarъ (“magic, sorcery”), *čara (“magic, sorcery”)" [wiktionary] to compare karma and charm, Latin carmen, Old Latin casmen "song", ety uncertain (*kenh2?); if the s seems to contradict the attempt, assume *carsmen and move on to Celtic. c) How many Sanskrit words in bhr- do you know? I know Brahmin, assume "scribe"(?), actually "high, lofty". That's *bʰerǵʰ-, which lists PCelt *brixtu, that is in Matasovich as "magical formula, incantation", indeed, Breton brith, PIE *bʰerǵʰ- "enlighten ?". But wait, it get's better. Your link lists PCelt Celtic *kʷritus
"Old Irish: cruth (“form”)
Welsh: pryd (“form, time”), prydaf (“compose poetry, versify”)". Matasovich has Celtic: *kʷritus
"magical incantation, form".
There's further "word", "judgement", under *b- *br-, but also "robber" (German Einbrecher?), a word "drinking cup", pela-[gasi?], Ger. gebrauchen "use", prepare, ...
It seems the cognate under Celtic: *kʷritus do not quite allow the given interpretation, but I will take it for the time being. What a fantastic journey. Thanks for the invitation.