Op, I have the same pain, although mine never goes.... and twins who were breastfed as well (well into toddler hood).
I have hypermobile joint syndrom/elhor danlos sydrom and chronic pain so have more than tried everything suggested here and more. There are a bunch of obvious things missing from my pov, so here goes.
If you havn't had experience of pain related issues before its important to start by treating it as medical, and going to the gp, repeatedly if necessary and to a different gp if you don't feel listened to. Osteos/chrios/acupuncturists while they may have a valid place (mine picked up on my hypermobility after a decade of physios/consultants missed it) are not medical professionals. They may have a great deal of knowledge and training (but again may not) but they do not have the medical back ground a hcp has.
If gp has ruled out the likelyhood of galstones/kideny stones or other medical possibilities- are these are looked at on the likely 'risk' of you having these not a 100% certainty at initial stages- they refer onto physio. Physios will treat as msk, but if there is not a reasonable improvement then they can recommend mri and further specialists for assessments. From a msk pov you are over weight, have been through a twin pregnancy with all the relaxin that brings, lost alot of core strength and muscle tone, and likely changed your activities.
Consider what you did before and what you do now from a msk pov- you carry two kids, you bend to put them in buggy/car/cots etc, do you also spend a lot of time in the kitchen? Bending over counter tops, to prepare foods/clean up? This would hit that exact point in your back, combined with heavier breasts/perhaps bra not fitting great, and add to the stiffening up many pepole begin to get overnight that would likely explain it. Pilates is the safest way to beging to build core strength but look for a pilates studio before some fitness pilates in a gym- post natal pilates/back care pilates/private sessions, and practice in between along with loosing weight. Physio can check if your stomach muscles have split, advised on posture, foot wear (podiatrist referall also), provide specific exercises. This is the starting point, if you don't have improvement or it gets worse you go back to gp.
You must take pain killers as gp advises. Chronic pain begins as regular pain that we become sensitised to because when our body experiences pain and the nerve endings split and the receptors in our brain increase to pick up the pain signals, which then means we experience a lesser pain as worse. Taking pain killers interrupts the pain signals so we don't become sensitised. Paracetamol can be taken with a NSAI like nurofen, but you can also have a low dose opiate like cocodamol or copying dydramol (which also contain paracetamol) ontop of nsai's, and there are stronger nsai's also. The breast feeding support network can advise on what's safe to bf on, their specialist pharmacist allways knows more than GPS on this subject and can provide research for you to print off and take in. Painkillers should be taken even if you don't feel then pain during the day as they prevent it.
Breastfeeding is a red herring btw- the research shows bf dosen't increase the relaxin that looses joints, in fact because it often stops periods it prevents the small amount of relaxin that kicks in to start our periods and it stops the cycle that drops the (stabilising) oestrogen and ups the (destabilising) progesterone. However, the positions we end up in, especially with twins, can lead to back pain. Also, you say 20 months? Have your periods returned? The restating of the above hormones could make joints looser. When we experience pain our core muscles switch off which then increases poor movements and further pain. Strong core muscles protect us from pain, hence why pilates works for so many.
Also, look at diet- magnesium and iron deficiencues, along with vit d like suggested can cause pain, although more commonly achy all over pain, but after everything your body has been through and how easy it is to ignore ourselves when we have twins, have a look at feriglobin plus liquid and a narural magnesium liquid supplement.
Chiros (I would never touch them) oesteos (mine were amazing, and used to be physios) acupuncture (NHS pain clinics offer electroaccupunture that has peer reviewed evidence basis), bowen techniqe/Alexander technique all worth trying by thus stage. And the pain clinic will recommend if you have been through every other option by that point.
Tens machine, heat pads or cold packs to reduce swelling (whichever helps you), magnesium salt hot baths, magnesium spray to apply directly. Massage, aromatherapy, infra red saunas. The mox box acupuncturists use is lovely imo.
Building muscle takes along along time, and needs a lot of practice small daily physio exercises daily, pilates, hypdrotherapy, take real commitment to produce successful results of reducing pain- but it manages it long term.
Hope this isn't too much of an essay for you ; ) hope your twins are better soon. Oh and invest in soft structured carriers for carrying, or wraps or slings- it's safer on your back than just lifting one on each hip. My 7 yrold still take a turn in our preschooler connecta.