Tula LP primer cups are slightly larger than Winchester/CCI cups and require more effort to seat them. Tula SP/LP primers are more troublesome to seat in the Pro 1000 and with some once-fired or tighter primer pocket cases like S&B/RWS, Tula primers won't seat fully and flatten above the case base.
For me, Winchester/CCI primers have worked well with Pro 1000 primer feed attachments and Magtech primers have fed the best.
In the current component shortage, I would obtain whatever primers I can get at the lowest prices. It's much cheaper to uniform the primer pockets.