My reading is that any amount not paid in 60 days is subject to $35 plus 2.99% (or another $35)...
If the amount due is not credited to your account by the 19th day after the closing date of the next billing period (approximately 50 days), we may assess a fee of $35 (Iowa $15).
If a payment of the amount due is not credited by the next closing date, we may assess a further fee in that same billing period of the greater of $35 or 2.99% of the amount due that was not credited by that next closing date (Iowa $15).
A late fee is assessed only on the outstanding balance, not on any new charges
Correct - it's a "late fee" as you said. Not "interest" like the OP mentioned.
As it is a charge card, there is no APR and hence no interest (monthly or otherwise), but rather a monetary penalty reflecting the size of the missed payment with a minimum flat fee of $35. Similar to an overseas ATM fee which is (e.g.) $3 or 3% of the withdrawl amount, whichever is higher (but not "interest of 3%" on an overseas withdrawl)