Travel from Brussels to Bordeaux

I am thoroughly confused here. Are you starting from Brussels airport or from the city centre?

I do not know what a "TVG" is, I think you mean "TGV" which is a French High Speed train. The trains from Brussels to Paris start from the centre of Brussels (Bruxelles-Midi station) and are branded "Thalys" or "TGV", those from Paris to Bordeaux are branded "TGV".

From Brussels city centre you have to go via Paris, then cross Paris (by métro) from Gare du Nord to Gare de Montparnasse, which are two different railway stations in Paris. You then get a train from Gare de Montparnasse to Bordeaux.
From Brussels airport, you would first have to take a train to Bruxelles-Midi. These run regularly. You then get the train to Paris. For example:

Bruxelles-Nat.-Aéroport depart 13:57, Bruxelles-Midi arrive 14:17
Bruxelles-Midi depart 14:37, Paris Nord arrive 16:02
Paris Montparnasse depart 17:20, Bordeaux-St-Jean arrive 20:42

Look up times on www.bahn.de but you cannot book this trip on this site.

Thalys has the monopoly to Gare du Nord. The TGV runs to CDG airport. There are only a couple of tracks at the airport, so it's easy to connect. How short is a short time? Just be standing near the door between the cars with your stuff as the train pulls into the station. If first class isn't much more expensive, it will make the transfer easier as you'll have fewer people jocking for position in your cars. Have you checked for flights between Brussels and Bordeaux on skyscanner.com?

I am sorry, typo - I meant TGV. The times listed for transfer at the airport TGV station are 18 mins and 23 mins depending on which time one leaves. When going from the airport station to Montparnesse station the transfer time is about 1 hour and 18 minutes.

I would definitely choose the departure with one connection at Charles de Gaulle TGV station instead of making an extra connection across central Paris by local transport. 20+ minutes to connect in one station should be fine and those high-speed trains do offer some protection in case of train delays. Try not to be the last one off the arriving train, however.

If I read between the lines, it looks like you want to get from Brussels to Bordeaux without doing the cross-town transfer in Paris from Nord to Montparnasse.
1. Take the 8:16 Thalys from Brussels Midi to Lille (Europe) arriving 8:50 and change to the TGV to Bordeaux departing at 9:13 and arriving Bordeaux at 14:37.
2. Take the 12:06 TGV from Brussels Midi to CDG Aeroport arriving 13:48 and change to the TGV to Bordeaux departing at 14:09 and arriving Bordeaux at 18:37.
3. Take the 16:00 TGV from Brussels Midi to CDG Aeroport arriving 17:53 and change to the TGV to Bordeaux departing at 18:09 and arriving Bordeaux at 22:37.
You may have to go through www.capitainetrain.com to get these routings.

Thank you for your replies. It looks like the first time listed by Sam is the best option for arriving in Bordeaux at a reasonable time to pick up a rental car. Thanks again.