I don’t think it’s required to let them know beforehand, especially if you don’t have an offer for the promotion yet. Whether or not you should out of courtesy or forewarning, I’m not sure. I think it depends on how supportive your SM truly is of your development as a future supervisor, even if it means losing you indefinitely.

If they wanted to keep you they would give you the hours to stay

I'd actually advise against applying directly, even though it's a promotion it'd still be a transfer, which you need to talk to your sm about first and foremost (*especially* if you're looking at stores in the same district.) It sounds like you have a good relationship with your sm so I'd sit down with your sm and be honest, you want to move up to an ssv position, even if it means moving to another store. As much as your sm may not want you to leave if she truly appreciates you she'll support you and put a good word in with the sms who run the stores you're looking at.

Regardless of her support though it's a really bad look to go behind her back. Even if she tries to block you, having that conversation is what matters. Your sm is not allowed to prevent you from leaving, and if she tries that's when you talk to your dm and/or the sm of the store you're interested in.