Being a new mom is hard enough, without bipolar & milk supply stresses. Can you get some help to come in during the day so you can focus on breastfeeding for a week or two? Putting in a major effort now, while your body is still primed to increase supply, is likely to be easier and more effective than making small efforts over a longer time. Are you being treated for bipolar? There are treatment options that are compatable w/ breastfeeding. Remember that a healthy mom is the most important thing, so if you do end up needing some of the meds that are not compatable w/ breastfeeding please don't beat yourself up over it.
Pumping is often recommended to increase supply. Ideally, you would pump immediately after breastfeeding Noah at as many feedings as possible. The goal isn't to get a lot of milk (you shouldn't; if you do that is a likely indicator that he's not nursing effectively), the goal is to signal your body to up production by increasing stimulation. Whenever you can pump after feeding, do so. No matter when you pump, even if you pump and then Noah needs to nurse, just go ahead and feed him. Your pump isn't as effective as baby (assuming he's doing a good job of removing milk) and there will still be milk available for him. You might find that he wants to nurse on each side more than once, or nurse for longer than usual, just follow his lead.
Breast compression is a great idea. Jack Newman's videos have some great demonstrations of this, as well as what signs to watch for to tell if he's actively nursing or just hanging out and nibbling. Also, make sure he's latching really really well with a wide open mouth, a deep latch, and lots of breast tissue, especially his lower jaw. Some of the "laid-back" breastfeeding positions (google "biological nursing") are helpful for this.
I would second Megan's comments about looking into domperidone. But remember that no medication is going to be effective without adequate stimulation. I really think getting some in-person help with an IBCLC experienced w/ milk supply issues is well worth the effort. I don't want to upset or offend anyone, but sometimes hospital-based IBCLCs are certified because their job requires it and are only used to working with brand new babies, where the goal is simply to get them latching on. The plans and recommendations for a mom several weeks postpartum with ongoing bf issues are different. Many IBCLCs do house calls, and in fact it's likely well worth the cost to have someone come to your home because she can help you with latch and positioning with the chairs and in the environment where you'll be nursing most of the time. Your insurance may cover this, it may not, but it's much much cheaper than formula and well worth the cost in the long run. You can go to the ILCA website to search for a local IBCLC, or call your local LLL Leader, who can likely give you names and numbers of local IBCLCs. The hospital where you gave birth may also have names and numbers for private IBCLCs.
The website I mentioned before, lowmilksupply.org, has tips and ideas about offering supplements in ways that are less likely to impact breastfeeding. Your LO may be screaming at the breast because he's become used to the fast, effortless milk flow from a bottle. Eliminating bottles if at all possible, or choosing bottles that are less likely to cause nipple preference, is really a good idea. A nursing supplementer might help but if he's not latching well and not suckling effectively it is unlikely to "teach" him. Again, in-person help. . . It can also help to do some breast compression, nipple massage, or even pumping to get milk flow started before offering to nurse. Using a medicine dropper or syringe to drip milk or formula onto your nipple as he's latching can help encourage him to latch if he's reluctant.
1 oz in 20 minutes, depending on his age at the visit, might have been an okay intake (we'd expect about an ounce or two at a week old, 2-3 per feed if he were 2-3 weeks old, but not every feed will be the same amount). The second test weight is more concerning. If he's not effectively transferring milk, then you really need to pump frequently to provide your body with adequate stimulation. At a month old, we'd expect him to need about 25-35 oz of milk a day; it sounds like he's getting the majority of that from formula. Hanging out at the breast, falling asleep at the breast, and "nibbling" are not adequate to stimulate your body to make milk. Milk removal is necessary. If he's really not effective at the breast, then it may be a more productive use of your time to feed him via an alternative method, rent a hospital-grade pump, and really really focus on pumping to increase supply for a few days, then work with someone to help Noah latch well and nurse effectively (which will be easier for him to learn if there is milk there!).