Tzitzit and Shaatnez

We know from the gemara in Yevamot that Tzitzit pushes away Shaatnez - and this is the classic case of עשה דוחה לא תעשה - a possitive commandment pushing off a negative commandment.
However, that's only if you had a garment made of linen with tzitzit made of wool, but if the wool and linen were woven into the garment itself, the problem of shaatnez remains.
Unfortunately, recently, hundreds of Talitot of Hassidim were checked and some were found to have Shaatnez, reports Ladaat.net. In addition, garments that were supposed to be 100% wool were found to have synthetic material. The Belzer Beit Din gave rules as to what items are refundable.