Yes, you can travel with regional trains. It takes a little over 4 hours plus the connection times, and there are no good connections, so count on 5 to 6 hours for the journey. The bus takes 4–5 hours and is cheaper than regional trains.
It isn't easy to find the train times or prices because online planners tend to go for the fastest journey only. There's an option to avoid ICE on the German railways site but that only avoids German high-speed trains, not others such as Thalys. Try looking up Bruxelles–Lille, Lille–Amiens and Amiens–Paris separately.
As of today, you can break your journey as follows:
- Paris–Amiens: 21.60€, lasts 66min to 99min, about 1 train per hour.
- Amiens-Lille Flandres: 21.10€, lasts 80min, about 1 train per hour.
- Lille Flandres–Bruxelles with a change at Tournai: 22.60€, lasts 102min, about 1 train per hour.
That's a total of 65.30€. It should be a little less if you go to a train station and buy a single ticket (I don't think you can buy that ticket online).
In comparison, a last minute Thalys ticket costs 99€ and the journey is only 82min. If I elect to travel one day later I find seats for 69€ (obviously this depends on how full the trains are). A last minute ticket on Idbus costs 39€ for a 4-hour journey or 29€ for a 5-hour journey, with no break of journey either way. So regional trains don't seem like a good compromise between price, comfort and speed.