TG will not pay for your hotel and certainly not any fees for changing your connecting flights on a separate ticket.
If you call TG, TG will most likely refer you to BMI. However, BMI will not be able to rebook you unless the flight has availability in I class. You will certainly not be rebooked to a non*A partner.
In my opinion, rather than spending a lot of time on the phone, being referred from TG to BMI and perhaps back to TG again, I would book a hotel on my own downtown BKK and spent the night there. Hotels in BKK are cheap so your loss will be minor and you will be able to enjoy the evening/night in BKK. You can also choose to stay at the Airport Novotel , but that can be quite expensive. There are cheaper hotels near the airport - try to search in the TG and Thailand forum.
Your flight leaves quite early in the morning. If you prefer to fly later in the day call TG; your request will most likely be accommodated if you explain that this flight is inconvenient and too early for you.