Sorry but honestly with dogs the last choice would be carpet. I have seen many beautiful old homes where wood floor was replaced and to help in the transition the floor was layed in the opposite direction it was finished to match and honestly looks fine. . If you must why not nice wood floor done to match and a large carpet that at least can be cleaned properly. A hard twist or polypropylene it is soft on the feet and easily cleaned

I’d go for wood floor running in two opposite direction instead of a rug. Especiay where dogs are concerned. Where does the transition occur from one direction to the other and can you put an area rug over the transition?

Loop has been recommended in the past. But there may be an issue with the nails of your dogs snagging larger loops. If the loop is dense, it may noy be an issue. Heavier face weight should help. Nylon. Wool nylon carpet is common in Britain where dogs are often part of households. Stanton is one source.

Do not do carpeting because of wood floor transitions going in different directions. Tell me what is wrong with these. Nothing.