I've had best results using cobalt bits, moving slowly through metal that's kept cool enough (either oil, water, or cutting fluid). As mentioned already, going too fast, without lube
will cause it to case harden. You'll then need to remove some material in order to get past that hardened area. Or just move to another spot for the pump mounts. If you can get someone to help you, try running water (slowly) over the piece you're drilling in order to keep it cool. Or, get someone to keep adding oil as you drill. I also make pilot holes first (small bit) then step it up to the size needed. You can sometimes use a step bit, but [IME] that's more for larger holes than you probably need. Also, changing the bits gives the piece some time to cool between steps.
Personally, if at all possible, I like to use a drill press for going through stainless whenever possible. I set it to go as slow as possible, use a cobalt bit, keep the pressure what it needs to be, and use lube.
Something you could try, if you're brave and have the tools needed, is to anneal the stainless if it's been hardened. That can be a tricky process, depending on what you have for equipment.