National Express allow you to pre-book an extra suitcase for an additional fee. Yes, it's an annoyance, but at least you will get all your luggage onto the coach, guaranteed.
We have travelled London-> Southampton several times with 20KG suitcase, roll-on and backpack each and never yet encountered any issues, although I do tend to pay extra for the roll-on, 'just in case'.
https://www.journeyintodarkness.co.uk/national-express-to-southampton-cruise-terminal