Dear Bertrand&Daniel - could you check the following? I neither speak French nor Math :-) but I computed the same skein relations with magic and trickery (so my results are unproven, of course). I get 11+6+28+23=39 basis 3+3 tangles (which I here already split into the D6h symmetry classes) and under the E7 family polynome 10+5+27+23=35 are independent. The linear rest should be your skein relations. So, are there exactly 4 (including symmetry-transformed!) of them, i.e. your Figs. 2/3? Since they don't look symmetric under rotation, it's hard to check for me.
(BTW, for 2+2 tangles, I "know" an even generalized result (of your Fig. 1) for 20 years :-)
Maybe going to 4+4 tangles could solve the problem of having a complete reducing skein set (or at least give an unproven solution) but since that might need hundreds of basis tangles you a) either have to crowdsource the calculation or b) do it with birdtracks or c) use math (I'm out then :-)