Some free advice. If I was Santa, I'd work it this way. Choose two people to be the doctor and detective. Tell the person you chose that you are Santa and they are to protect you every night. You'll keep choosing them over and over. Next, you tell your investigator who you are and who your doctor is. That way, if you have selected a mafia member to protect you and they do not and whack you instead, well, it sucks but it exposes a mafia member as the detective will know who failed to give protection.
Meanwhile, the detective must give you all his findings. From there, you will start to build a townie alliance and keep those people informed of each new investigation for each day. The key is to find Santa's Helper. Once you've found him, he's the only other townie you can fully trust so make him your detective from then on.
What if the person you choose as detective is mafia? That's a risk but he will be in a quandry as well. He can give you false readings. However, doing so means he may also be caught as well. As soon as someone dies who isn't what he said they were, then he is exposed. He'll die and you'll start anew.
However, a smarter thing might be to rotate. So, the investigator you choose on Day 1 will then be your doctor Day 2 and you pick a new investigator. Then your Day 2 investigator becomes your doctor on Day 3 and you choose a new investigator. Of course, you would tell your investigator who you want them to check on and start accumulating data that way and keep it from all being potentially corrupted.
Of course, if both people you choose are mafia members from the start, well, that is some really crappy luck but there is always Santa's Helper to take over and hopefully things go better than. Plus, there will be players winning investigations each day and so things can still work in the townies favor even if the Santa role is crippled.
It is the best plan I can come up with regarding these roles.