Maybe as simple as a smaller pulley on the existing motor.
The pulley is pretty small already.
About 25mm diameter cogged pulley on a 9 mm shaft.
My first shot at rigidity would be less stickout on the tool and the part, then I'd look at a live center in the tailstock, and/or a follow rest.
Definitely those would all be good things to address.
7x lathes suffer from looseness in the gib/dovetail on the compound and cross slide, and also poor attachment of the carriage to the ways.
Some of these things aren't such a problem with even a bit larger lathe, where added mass adds inertia.
It's instructive to put a steel bar in the toolholder and give it 'a wiggle'.
For many 7x owners, it's their first lathe so analyzing problems is difficult.
Also, our techniques can be pretty poor!
I see a lot of 7x lathe pictures with the compound still in the 'shipping position', parallel to the ways. Users get the compound 'hanging out' toward the headstock, and that doesn't help with the chatter and looseness.
But if it's so common a problem I bet there are other members here who are better experienced with what works and what doesn't.
Yes, they are probably hiding here somewhere!
That's why I recommend people to go where the other 7x sufferers hang out.