Flower Duet, it is no problem for you to take a suitcase to training with you. A lot of crew bring luggage when doing recurrent training, so it's not out of the ordinary.
I used to treat myself to a room at the Meridien in the north terminal (I believe it's called something else now, arora perhaps?) and i used to pay around £70. Not bad for a night. If you are after longer term accommodation during training, you can get a BnB for around £30. You will need to search the accommodation threads here and cabincrew.com
cheers