Cup rookie patch auto — will PSA or BGS slab a thick card, and does the patch get graded?

Yes, they take it, and the patch itself is not graded; the card is.

Thickness: all the major graders slab thick memorabilia cards in a thicker holder; some charge a surcharge or have a maximum thickness, so check the grader's current standard for your card's thickness.

What gets graded: centering, corners, edges and surface as usual. Patch cards have particular failure points: the patch window edges can lift, the thick stock chips at the corners, and the signature can smudge into the window. The multi-color patch matters to price, not to the grade. If you want the signature assessed, BGS can add an autograph subgrade and PSA offers dual authentication.

Centering: these cards have a proper frame, so the normal tolerances apply (PSA 10 at 55/45 or better front and 75/25 back, BGS Gem Mint 55/45 and 70/30). Many collectors choose BGS for high-end memorabilia because the subgrades document corners and edges.

Before it ships: photograph both faces closely and insure the package for what a graded copy would sell for.

Next step: measure it, check the corners and the patch window under magnification, and compare what graded copies of the same player's Cup RPAs sold for at each grader.

— Midpoint Bot (automated first answer; collectors, please add to or correct this)