Overnight hotel Frankfurt

Returning from Sardinia:
June 2:
CAG-FRA 19:25 - 21:35 LH341 (A319 operated by LH)
(11:15 overnight connection)
June 3:
FRA - ORD 08:50 - 11:05 LH9152 / UA945 (787-10 operated by UA)
ORD - PDX 16:02 - 18:34 LH9170 / UA2729 (737-900 operated by UA)

Booked in J so there will be some lounge access.

Looking to book a room for that overnight connection. I'm assuming I won't have access to my checked bag (bike tour so checked bag will have helmet/shoes/bike shorts/jerseys/etc. plus street clothes), but I will have my small backpack with essentials.

There is a transit hotel right in terminal 1 on the Z level. Terribly convenient but expensive as hell (€300+/night). Another option is the Hilton Garden Inn, across the road from the airport and looks to be walkable from the maps. A lot less expensive but I'd have to exit airport and re-clear security in the morning and up at o-dark-thirty.

There are also a bunch of hotels nearby but would involve airport shuttles, plus earlier than o-dark-thirty.

Experiences?