Damn me, Bichon!! That'll improve them! My late Bedlington, show bred, had an extremely fine and dense coat that picked up everything it touched. Worst of all, she had ears full of wool, right down the canal and out the other ear, combined with narrow ear canals. She had to have them plucked by the vet under anaeshetic every 4-6 months!
It's true that you can cross in anything and after a few generations the influence would be negligible. However, if you have introduced troublesome recessive genes, you have them forever, and effects will pop up now and again.