There are a couple of RAW reasons to prevent Multiclassing
Multiclassing itself is not a required rule and is listed under Chapter 6 of the PHB. The relevant statement can be found on page 163:
Your DM decides whether these options [multiclassing and feats] are available in a campaign.
In addition, in order to Multiclass to Druid, there are prerequisites that must be met for both current class statistics and new class (page 163):
Druid Wisdom 13
Rogue Dexterity 13
In the case of the character cited in the question with a Wisdom of 9 and Dexterity of 13, they would be unable to multiclass into Druid even if Multiclassing was permitted by the DM because of failing to meet the required Wisdom prerequisite (but they do meet the Dexterity.)
Are you having fun?
Ultimately, it sounds like you are having some frustrations with your DM (arbitrary limitations, only XP for kills, etc.) You should ask yourself are all of these things getting in the way of you and your son enjoying the table and the game. If the answer is yes, then I'd suggest trying to talk to the DM and if there is no positive resolution then walk away from the table. Making this a bad experience for your son is not going to do much to have him continue playing.
However, if the rest of the sessions and the group are enjoyable and these are more frustrating bumps in the road that don't overshadow the good times, then it's just something that you'll need to accept (but you can always complain!)
Starting Over at Level 1
Yes, your son will be a significant disadvantage joining the game as a Level 1 character in order to begin anew as a Druid. However, you are only looking at about a 3000XP difference. He'll get to level 3 pretty quickly (as long as you help keep him alive) and will only be behind for a bit after that. As the levels get higher, the difference is much more negligible. It may be worth it to get your son the character he wants to play.