They Can Try:
I think this might be more of a writing SE question and less of a worldbuilding one. I'm on both stack-exchange sites, so I'll try to hit both sides. It all has to do with the psychology of your character and the physiology you choose to have the vampire deal with. So there is no RIGHT answer, per se, but that's a philosophical view. An "absolute" right or wrong answer might be better addressed by an Imam, though.
I have had vampiric characters who desperately tried to cling to their belief systems, to the point they went to a Catholic Church are received communion (and the magic/faith system of the game had interesting things to say about that...) A good book to read that deals with something somewhat similar is When Gravity Fails, a scifi novel about a Muslim character who doesn't have much faith until after he gets cybernetic implants that are technically considered against the faith. Ironically something that made him more out of synch with his religion helped him find belief.
The struggle to be faithful in the face of a reality that makes it hard is a beautiful literary trope. In my mind, it is less critical if the character is technically following all the rules, and more important HOW he deals with the choices, and how he reconciles the seemingly impossible paradoxes of his new life. This sounds like the makings of a compelling plot element.
- I believe Muslims are still allowed to receive transfusions (but I'm not Muslim, so I'm not 100% - on the other hand I know Muslims who work as blood bankers at hospitals) since medicine is in a separate category. Some faiths DON'T even allow this (Jehovah's witnesses, for example). Perhaps your character gets transfusions instead of eating - so then they aren't violating the letter of the restriction.
- Perhaps there is a VampiricMuslim sect of vampires, and they have their own Imams and rules, making VERY different interpretations of the Koran to fill their needs. In this case, being a practicing Muslim becomes more definitional again - if the subgroup declares blood is essential, the character can decide if they accept the group ruling or not.
- Your original interpretation about blood might not be completely off base. There is controversy as to whether foods that contain SOME traces of blood are acceptable or not. It looks like there may be enough wiggle room to allow a Muslim vampire who interprets things the right way to at least consider themselves practicing. The strictest interpretation would preclude things like raw beef, but not all.