Adding any fish to a betta tank is for your benefit, not the bettas. Your betta will be perfectly happy on its own as long as the aquascape in your tank keeps the betta engaged. Keeping any fish with a betta is a risk, and it will be down to the temperament of the individual fish as to whether it works or not.
10g is on the small size to keep any group of schooling fish. You might get away with some kind of small rasbora or tetra, but they would be better in a larger tank. A larger tank would also reduce the risk of your fish clashing personality wise as they would have more space to keep out of each others way. Neon tetras, galaxy rasbora (celestial pearl danio), corys and harlequin rasboras are schooling fish commonly kept with bettas. A couple of platys might be a better choice in 10g, but only 2 or 3 fish. Avoid anything with long fins as the betta can mistake then for other bettas.
If you proceed with your plan, get your new tank set up and stocked with the new fish first. Cycle the tank and let them settle in before adding the betta. This way you wont be introducing the new fish into a tank where the betta has already decided it all belongs to him. This will reduce the risk. Keep your 3g tank in case you need to remove the betta. He might react with extreme violence immediately, after a few days, weeks, months, years. Or maybe never. But have a plan ready should you need it.