The bishop has not placed Catholic politicians in any conflict except with their own conscience. A Catholic politician can serve in a legislature in good faith. But according to the bishop, he cannot support abortion. This would mean that a Catholic politician would have to vote against public payment for abortion (not required by the Constitution), he would have to publicly aver that he personally is against abortion, and he would have to support changing the Constitution to forbid abortion.
considering that lawmakers don't interpret the Constitution anyway
Lawmakers interpret the Constitution every time they vote for a bill, because they're supposed to be judging whether or not that bill is consistent with the Constitution. Unless their objective is to either subvert the Constitution, or to vote on the basis of political advantage and let his fellow legislators or one of the other two branches of government take the heat.
Bishop Burke is reported to have been asked what will happen if his position disqualifies Catholics from running for office.
A bogus question. Nothing in his position disqualifies Catrholics from running for office.
Exactly. These are all reasonable things - and, incidentally, reflect positions held by many non-Catholics as well - and none of them conflicts with the Constitution or in any way prevents a Catholic from holding public office.
Well, at least until the time comes when the pro-abortion contingent makes the personal performance of an abortion to be a requirement for anyone who wishes to run for office in the US - something I think they would dearly like to do (if they're Dems, at least).