I know that compromise sounds initially like an attempt to square the circle. I don't think it means abandoning, or diluting the values that each side holds dear, but tolerating them side by side, accepting the sincerity of the other side's convictions, finding peaceful ways of championing and pursuing them - and giving each side the space to follow their own values as far as is possible peacefully.
The alternative leads all too easily to the sort of horror that we had in Northern Ireland, which blighted the lives of people for a generation and spilled over into the wider UK and beyond - the atrocities and violence that destroyed local democracy, killed thousands and ruined the lives of hundreds of thousands. It was only when the opposing sides decided to find a compromise political framework and tolerate each other that a stable peace was established. A key part of this was that they stopped seeing each other as the devil incarnate and decided to work together to run their community. It's not perfect because the conflicting values run deep, but it's a million light years better than before. This wasn't at the cost of either side abandoning their dearly held convictions - though they have been less vehemently held as the years go by and as younger generations take over from the old - but they do have to accept that the other side has the right to follow their own star unless it destroys the peace again.
I guess there are examples of other political systems in the world that solve the problem in different ways. For example, there are theocracies, and near theocracies, particularly in the Moslem world, that pretty well dictate the values that people are expected to commit to. The states like China and North Korea are very similar but are rooted in (pseudo?) communist-related values rather than religious ones. These all use force to make people conform, and the consequences of dissent on the fundamental issues are pretty unpleasant. This is certainly a way to resolve the issue for the lucky faction that holds absolute control, but personally I think it's better to find a way for groups with differing values to live together in a tolerant, free democracy. I think it's the least ethically indefensible of the alternatives.
Like you, I find this is a hard thing to handle in a way, being brought up from childhood as a Roman Catholic - from pre-Vatican II days as well, when the Church was far more dogmatic, indoctrination-al and introspective than it is now. I think a good way forward for anyone with these sort of values is to 'hate the sin but love the sinner'. This works both ways too, and I'd hope for the same tolerance from people who believe that Christianity brings great evil, not good.
Mind you we could always try and bring the known world under a Roman dictatorship again