I understand the need to keep the harness on when housebreaking, I went through the same thing with Lola and I also use the bell.
Scrap those skimpy strap harnesses, I have always hated them, they are confusing to put on, hard to adjust and seem to be uncomfortable. Below are a few suggestions since they are both more like clothing.
The first style is from a gal I have bought over 10 harness dresses from, I have both fancy to plain styles but they are all made the same, with tons of strong velcro around the neck and tummy and a secure reinforced d ring strap that runs down the center of the vest. Lola wears these almost exclusively when we go on walks and she is 5.25 Lbs and a very aggressive walker that will pull to chase a squirrel or bird. I find them secure and safe enough to lift her off the ground if I have to. They are no different than keeping your dog in a sweater or a dress when around the house. Plus they are very affordable and you have a lot of color choices, Karen makes them to fit your dog and are easily adjustable.
The second and third style is even more secure as it closes in the back with velcro and then a double d ring latch, not bad to keep on but probably not as comfortable or stylish as the first one. Since Lola only needs to go out about 3 times a day it's what I put her in to go potty and I keep it on a retractable leash by the front door. The one I have is from Wallmart and sells for around $10.00 but I am not sure that the XS will be small enough for your puppy at the moment. I am sure one of these styles will help solve your problem of keeping the harness ready so you can get your dog out in a hurry.
Dog Bones Harness Vest with Collar by graciespawprints on Etsy 
Amazon.com: Puppia Dog Mesh Harness "Vest Soft"... SimplyDog Mesh Dog Body Harness, Black, Multiple Sizes Available: Dogs : Walmart.com