According to www.jizdenka.cz Trier to Bourg-en-Bresse is approx. 64 EUR.
I think you might be able do this cheaper if you travel on the German side of the Rhine. There are special tickets that are valid in the individual Länder (federal states) in Germany that allow you unlimited travel on regional trains. The most interesting ticket for you would be the Baden-Wurttemberg Single Ticket or the Rhineland-Palatinate Single Ticket.
You should also avoid the TGV in France, it is relatively expensive.
I played around with www.bahn.de for you, I found a connection using regional trains that goes Thu, 16.07.09:
(Trier Hbf dep 06:21
Saarbrücken Hbf arr 07:31)
Saarbrücken Hbf dep 07:45
Strasbourg arr 09:27
Strasbourg dep 10:13 4213
Bourg-en-Bresse arr 14:43
It is the only connection from Frankfurt Hahn to France that doesn't use the TGV.
With that connection I would catch the bus to Saarbrücken from Frankfurt Hahn (price 16 EUR), the bus leaves 00:30 and 04:40 from Hahn Airport. You need to prebook the bus, you can find more info here:
http://www.hahn-airport.de/
From Saarbrücken to Bourg en Bresse the train ticket should be around 61 EUR.
I did some more digging, but so far this seems to be the least complicated method of getting to your destination. Using the above mentioned tickets would involve going via Heidelberg and Basel, with a quick guesstimate you'd end up paying more and changing up to 10 times.