It seems to fit nicely...
I put on the 3 jaw chuck (without the bolts) and did the same. The slight movements and a little knocking sound was also there.
I measured the gauge with the caliper - 70.01mm. The registration boss on the spindle flange measures 69.99mm. That means that the recess is 0.02mm over sized... No wonder the gauge fits so tightly into the recess of the 4 jaw chuck. I've to press it in with my hands.
The work will still be completed. I'll decide if I need to redo the project after I test the completed chuck.