Bike Rental shops: As you can see from this list, they exist in many towns:
https://www.mosel.de/freizeit/radfahren/details/?tx_ttnews%5Btt_news%5D=241&cHash=8969334d08b6696084d7376790d2e143
I suppose it might be possible to arrive "bikeless" by train in a Mosel town like Bernkastel or Kröv or Traben-Trarbach for 2 nights, then rent and drop off in the same town before catching a train on to the next stop downstream (Cochem? Treis-Karden?) and biking that area for a couple of days.
I've rented bikes previously in Alf and in Cochem. I generally find 2-3 hours on a bike very enjoyable - and to be more than enough biking on the whole - so I've never done a multi-day tour like you're proposing. I know 1 or 2 others on this forum who have - maybe they will pipe in and share their experiences.
A new rental location in Alf popped up a few years ago where the ferry docks, just across the river from Bullay (a Mosel railway hub.) See photo:
https://img.oastatic.com/img2/13071361/834x417r/alf-faehre-blick-auf-bullay-.jpg
This is the main rail line for Trier > Koblenz with all train stops shown:
https://assets.static-bahn.de/dam/jcr:c3b41a1d-30c0-4d71-98b9-34f8fe093d50/190970-258769.pdf
Here you can see the offshoot rail line from Bullay to Traben-Trarbach:
http://www.alpenbahnen.net/assets/images/autogen/k-IMG_2375.png