With brand new Compomotive ML (minilite style) wheels, I had a hell of a game getting stick-on weights to do just that. The pastic coating surface was very waxy and needed a light roughening to assist keying.
Are you using stick on weights and are they not sticking on? - count em 😬
On a light car just a few grammes will make a difference.
At that time I also went to about half a dozen places trying to get wheels satisfacorily balanced. One made the car absolutely undriveable over 35 mph. Afterwards the man then said "Oh yeah mate, we've been having trouble with that machine. theyr'e supposed to come and fix it next week". So when you go to the tyre shop, ask when they last serviced and calibrated the computer balancing machine. Be very suspicious if significant weight is being applied diametrically opposite on one rim.
In addition to Hoopy's comments - is the rack on your car clamped in compliant bushes as per the Triumph type racks?
As a double check - I'd put a pair of 15"ers back on the front if you still have them and see if it still shakes on a quick experimental blat? If not - swap fronts for rears. This may save a lot of time if it show the wheels are the problem.