My approach would be to test a wide range of crosslinker concentrations, keeping everything else the same. Use SDS-PAGE to examine the results. The most meaningful results come from the low crosslinker concentrations. High concentrations are more likely to result in artifacts, including intramolecular crosslinking.
The next step would be to fix the crosslinker concentration at the low level found in the first step, then vary the protein concentration, which will help you get an idea about the dissociation constant of the complex.
If you are extracting the cells to run on SDS-PAGE, then you can use a denaturing buffer. You should use a reducing agent unless you are using a disulfide-linked crosslinker.
If you are going to run the samples directly on MS instead of extracting them from SDS-PAGE first, then you should avoid detergent. Most detergents are not compatible with MS, although there are a few that can be used.