Technically precision barrels do not require high quality BBs. They require BBs that are spherical in shape and smaller than the bore diameter. There is nothing wrong with low cost or low quality BBs so long as they are not defective BBs as in they possess physical defects like dents, cracks, seams, are not spherical, or are dirty.
To determine whether a BB can be shot through your barrel is to measure the BB and the bore with a caliper that reads out in 0.01mm or 0.001". Alternatively a micrometer may be used, although the extra precision isn't really needed.
If the BBs are of dubious quality, it may be wise to measure all of them, or random select a large number and check for any measuring larger than bore diameter.
Worst case the BB is larger than bore diameter, it will only get stuck, can be removed by cleaning rod or other rod like tool, and will not damage the barrel because plastic is softer than the steel or brass barrel.
Another potential issue with oversized BBs is safety in airsoft play. Larger BBs means tighter fit with bore, meaning less air rushes past the BB that results in higher velocity and higher energy that may be beyond field limits, especially if the gun passes chrono test with smaller BBs. The effect is increased further if the gun has a tight bore or a precision barrel installed, further reducing the gap between BB and barrel, and can result in significant increases in velocity and energy well beyond field limits. Different fields, locales and countries treat this problem differently. Some parts of the US they don't care, most of Europe will permanently ban you whether you knew or not.