Is it okay to run with a painful bruise + blisters on inside of heel?

You probably could gut it out, though it's likely not a good idea. It will probably slightly alter the way your foot hits the ground and that will carry along into other areas of your form and open you up to other injuries, not recommended based on personal experience.

Honestly, I would take a couple days off to let it heal. In the long run, taking 3-5 days off won't matter at all. If the shoe keeps rubbing on it, it will never heal and it will just start a frustrating cycle that won't allow you to get far in your training. Sorry, I've been there with achilles tendonitis. Cut the backs of my shoes off so my foot was moving around while I was running. Lots of bruising and blisters. These things heal quite quickly though. Doesn't sound like it's any injury other than just some surface stuff which is great!

You are not giving it time to heal.

Also, you may be altering your gait to accommodate and that could cause more serious problems.

Is it ok to run when injured?

sure buddy. Knock yourself out.