You are very wise to spend a couple days on the Rhine! It's great place to relax and settle in before your tour.
The problem is you are trying to book too early. September 2 and 3 2020, right, with Sept. 4 in Trier, right? There is NO reason at all for these small towns to be booked solid on Wednesday and Thursday 12 months out! That hotels are not yet bookable is the only explanation.
Anyway, don't get too carried away with the idea of staying in Bacharach yet. Have a look at the other options first. If you are looking for a town well known for its train noise and its lack of river-view hotels, Bacharach might be just right. Here's a photo of one of the Bacharach establishments Rick recommends.
Unfortunately, Bacharach is neither a convenient starting point or ending point for a Rhine cruise since it's right in the middle of this route, which stretches from Bingen to St. Goar:
http://www.rhein-zeitung.de/cms_media/module_img/1195/597502_1_arslideimg_597502_1_org_bacharach.jpg
Don't misunderstand - Bacharach has the most attractive old-world buildings on the Rhine and some nice places to eat as well, and no visit is really complete without a visit there.
But you can visit Bacharach in just a few minutes from other towns - like St. Goar, a very strong competitor for the visitor dollar (or Euro, heh,) especially for those using the trains to get around. Rheinfels Castle is right there in town, so you can tour it anytime you like (while Bacharach has no castle you can tour.) St. Goar also has a ferry crossing if you wish to explore the opposite river bank (think Rüdesheim and Marksburg Castle in Braubach, an intact medieval castle with daily scheduled tours in English.) As you saw in the ferry crossing above and here, the view across the river from St. Goar is fabulous. And you can enjoy this view from several waterfront hotels in St. Goar (Rheinhotel St. Goar, Hotel Rheinfels among them; some have balconies.)
St. Goar's location is near ideal for short outings - take the train south to visit Oberwesel (5 min.) and Bacharach (10 min,) or north about 10 minutes to Boppard. And when you cruise, take ONE train to Bingen, then the cruise ship back. No multiple train trips needed.
Boppard is also a great town with many similar advantages - and a good base town choice as well.