usually use RPMI-1640 with this cell line as following : RPMI-1640 tissue culture medium , fetal bovine serum (FBS), antibiotics (100 U/mL) penicillin, 0.1 g/L streptomycin) , Trypsin, HEPES, sodium pyruvate, and l-glutamine (powder).
I havent worked with BRIN BD11 cells but I have worked with media adaptation for cell lines.
DMEM is usually chosen for adherent cell lines whereas RPMI is used for suspension cultures. Also you need to think about your experimentation and the potential uses for the cells and the summary of your project. For example if you use your cell line for a fluorescence assay then Phenol red needs to be excluded from your culture medium as it has a high fluorescence and your background reading will be too high. All aspects are considered when deciding on the best media.
Further some cell lines also have the capacity to adapt growing in RPMI and DMEM medium. HeLa cell lines are one such. One big difference between RPMI and DMEM is the glucose concentration higher on DMEM medium. This would alter the glucose uptake by HeLa and these metabolic changes may lead to differences in signal events. When I have had to switch cells in different medium compositions, I sometimes take a gradual approach. For example if you are looking at switching from RPMI to DMEM, start out with 100% RPMI, then switch to 75% RPMI:25%DMEM, 50% RPMI:50% DMEM, and finally 100% DMEM.
I personally will research the cell line and see what is suggested and then try a few options to see what works the best for me. I usually go in for observations with respect to morphology of cells in each experimentation.