Hi, Phil. You cannot paint directly over fresh stain. Mineral spirits is the solvent for both of these but about any organic solvent will work as a solvent, too. So that means that you will have to put something that is not organic solvent based between the paint and the stain. The choice for this is alcohol based shellac like Zinnser's B-I-N.
Let the door dry with the stain on it for a few days. Longer is better but it sounds like you may want to install the door sooner so do let the stain set up and the solvent from the stain evaporate at least for 2 days. Then, you can apply a coat of B-I-N. You may get some bleeding. If you do, you can apply another coat of B-I-N, and another, and another, when each coat is dry. You will want to LIGHTLY sand between the coats and tack rag the dust. Don't sand through the B-I-N if you can help it and any place that has been sanded through MUST get another coat. Eventually, you will tell that the B-I-N no longer gets a tan look after a coat and you can then paint.
Good luck. You will be pushing it to do it this way and you will have to use some discretion as to when to change products.