To me the SPG & UR points have the most value in general so favoring them is a solid plan. The Hyatt card comes with $2k minimum spending to get the 2 free nights so that'd be a good way to get rid of that quickly just like the JetBlue card.
$2k on Hyatt
$1k on JetBlue card
$2-5k on SPG
Rest on CSP/CSR.
Did you talk with the hospital's billing department to make sure they're ok splitting over 4-5 cards? It's not something that is unheard of but it's best to clarify before going in with the plan. Especially if you're going to be sending Amex a chunk to increase the SPG card's limit.