Pull out both air/idle screws and blow out the passages with high pressure air. Point is to unblock any crap from the emulsion tubes. Fast, easy to do and may work to your advantage.
Next, swap on a known good carb and run it. Harder to do but it could help isolate on issue: is it carb related?
Very hard to do and a total PITA. Pull the tranny, remove the throwout bearing and arm, block up the engine at the pan and fire her up. If it still shakes, then yank the bell, remove the clutch and disc, refire it and see if your shake disappeared.
Story time! My late, great Uncle Bill bought a brand new black1957 Ford and love it. But, it had from Day 1 an odd vibration at various engine speeds. The dealer took the car in a few times but the local mechanics could not source the issue. Bill, being a stickler and standing by his rights (new car and all) pursued Ford with letters, etc. to fix the engine or give him a new one. Ford's Dearborn office did send a factory repr. as the dealer gave up. Bill took the car in on the appointed day and the factory guy pulled the engine, totally disassembled it and went to work. IIRC, it was two days later the call came in to Bill to visit the dealership. The rep, showed Bill all 8 connecting rods, pointing out the machined balance pads on each one's big end. One rod showed zero machining on its big end! He had taken all 8 to a local machine shop and had each one weighed. That one rod was way out of balance compared to the other 7. A touch of machining to match the others, a reassembly and the engine engine finally engine ran smoothly.
Hope yours hasn't the same issue!