Nope ... read POWELL v. McCORMACK 395 U.S. 486 (1969). This has been adjudicated before, albeit for the House but they didn't have the specificity of Amendment 17 to go by either.
The House of Representatives did this with an elected member back in the ‘70’s. They just voted not to seat him.
And SCOTUS overruled - read the case, esp. Justice Douglas' opinion. Seat him they must and then expel.
Imagine what happens if Burris then says "I ain't goin' no place---arrest me!" LOL!!
The “Adam Clayton Powell” seating in the House of Reps was the one I was thinking of. I had forgotten that that was overturned in the USSC.
On reflection you wouldn’t necessarily want a majority of the House arbitrarily refusing to seat a duly elected individual who was otherwise constitutionally-qualified.