I am not saying that it is the solution, but it’s a possibility. Check that the tyres are completely round.
We recently had the same problem on a Land Cruiser 79. The customer claimed a vibration on the left-front tyre, but there was nothing mechanical to be found. By coincidence we found out that the right front tyre was not round, and had a slight bump. With cold tyres, the problem is more evident, because the tyre is less flexible.
I advise you to go to the tyre shop, make a deal on a test with four new tyres, and if it is the correct solution to the problem, then you needed the rubber anyway. If it is not, put the old rubber back; and you are a lot wiser for a little bit of money.