Yes, you can use the Rheinland-Pfalz-Ticket for regional trains from Bingen to Koblenz to Cochem, with a stopover in Moselkern. You haven't said what days of the week these days are, but if your trip to Burg Eltz is on a Saturday or a Sunday, there is a bus from some of the stations on the Mosel to Burg Eltz. I think that bus will also be covered by the RL-P-Ticket. I'm not sure if there are public buses from the Cochem area to Hahn, but I'll check. I'm pretty sure the bus I mentioned (from Bullay to Hahn) is a private bus and wouldn't be covered.
Update: There are public buses from Bullay to Hahn, arriving at Hahn at 18:11 and 20:21. I don't know if you want to get to Hahn as early as 6:11 PM (almost 4 hours early) or as late as 8:21 PM (1h39m before flight time), but those look like your choices with transportation covered by the RL-P-Ticket.
The bus getting to the airport at 8:21 pm is the same one (VRM 750) I told you about earlier. It's a public bus run by the VRM (Verkehrverbund-Rhein-Mosel, the metro district for the Mosel and most of the middle Rhein). The times I first told you about were from the Hahn website and were probably out-of-date information. The 8:21 pm arrival time is from the current schedule on the VRM website.
In case you are going to Burg Eltz on a weekend, the bus that stops just up the hill from the castle leaves from several train stations on the Mosel. The last one is Hatzenport; it a 17 minute ride from there.
Another thought: There are regional Bahn buses, which are probably covered by the RL-P-Ticket, hourly from various points in Bingen, including the Stadtbahnhof and the Hauptbahnhof, to Hahn shown on the Bahn website. They take just under 2 hours, with a change in Simmern and get to Hahn Flughafen at 2 minutes past the hour.
So either leave your luggage at your hotel in Bingen after checkout and go to Mainz (or take you luggage with you to Mainz and leave it in a locker there), explore Mainz, and come back to Bingen for the 6:10 pm bus to Hahn (at 8:02 pm). Do the Mosel on day 2.