From my experience the cancer cell lines grows without any problems in all type of growth media (DMEM, RPMI, McCoys). We are using certain grwoth media because ATTC or other provider says so and everyone is publishing it after them. Media for cancer cells are slightly different in case of glucose concentration, amino acid content and so on.
Bicarbonate buffered media will only maintain pH 7.4 in the CO2 incubator, if you need to have correct pH outside of the incubator purchase HEPES buffered media!
The best you can do is to check your cell line on some cell line bank site like ATCC or DSMZ. There you'll find information about the specific needs of your cell of interest. Some cancer cells will do fine with DMEM but other will do better with some special media (e.g., RPMI 1640 is generally the best for blood cancer cell lines).