First of all, I will say, I am not a psychologist. However, I have experience dealing with populations who need emotional assistance, and neurodiverse populations.
My first comment is that you should encourage your partner to join this forum and find autonomy in seeking assistance for themselves. There is a huge component to TMS that makes a person feel powerless, and part of overcoming symptoms, dealing with personality and with inner and emotional life is feeling a sense of power over the challenges that we face. Does your partner even feel that this is a challenge, or are they totally fine with themselves. If they are fine with these issues you mention, than it's not their problem - it's yours. Something to think about.
If he is anxious and rageful, then perhaps re-reading Sarno, who talks about inner UNRECOGNIZED rage as the source of our pain (and symptoms which include anxiety and being stuck in mindsets like anger). Many types of psychological schools don't really touch on the subconscious, unconscious or even unrecognized rage and how all of these things feel in the body. They focus on how you THINK about those things, not really the emotions which are physical sensations in the body.
1) Journal - if your partner hasn't engaged in journaling then perhaps they might like to give it a try. Nicole Sach's journalspeak is a method of journaling which combines private emotional expression and then an emotional/mental calming and acceptance of those emotions. Here's how:
https://www.thecureforchronicpain.com/journalspeak (How to JournalSpeak — The Cure for Chronic Pain) They may even find resonance with her story, which includes back pain and the revelation that her inner, deepest emotional life made her brain think she was horribly angry and resentful at her children. This is of course not really true! She loves her children, but the anger and deep seated fear in her mind was that she didn't love them - it ties into her upbringing. Letting out all those emotions and all of the nonsensical "shit" that was deep down helped set her free. Does she occasional have symptoms...yes. But she has a method of which to suss out the source and they resolve.
2) Have them read Claire Weekes book "Hope and Help For Your Nerves" - it's a funny, old fashioned little book but it truly has the answers for dealing with anxiety written by a Doctor who experienced the symptoms herself.
3) Some people are resistant to the journaling. Their brains just can't get there. Currently, TMS practitioners suggest using EMDR, ISDTP or ISF as psychological tools. They all utilize the mind body connection and help the person connect feelings in the body that they may not be either recognizing or may be repressing and teach the patient to allow them to be physically expressed in the moment. This is simply emotional regulation. Each style focuses on different areas: EMDR uses the sensations of left and right as calming tools to allow the mind to better organize information so that it doesn't seem to present as being "unsafe" (eg. current moments that the brain relates to past moments). ISDTP allows the patient to recognize that most moments of our life and our emotions are physically represented and not to be feared. ISF takes a look at each and every part of our makeup and helps us accept ourselves and our feelings. Dr. Schubiner offers another method which is also said to be successful but as of yet, there isn't anything written about it - and I don't think they have a list of practitioners who use the method yet.
3) Take the ACES quiz and see if your partner can find even subtle instances in their childhood that may have created anxiety and trauma - from maximum to even minimal trauma. Take the test as if you are looking through the eyes of your childhood self. It is simply a tool to see where the reactions might come from.
5) Meditation: every person needs to find a way to chill out. Meditation separates our brain machinations from our true self and is settling - especially for someone with an anxious and busy mind. Just "being" without any kind need to think and fulfill our tenancies to things like perfectionism (which is partly what might be going on here...partner has a problem with accepting children who aren't perfectly behaving and somehow he feels this reflects upon himself). Making sure partner (and yourself!) have time to play and do things that don't have any relation to eg. self-worth, appearance, responsibility etc.
6) Just from what you describe, as a LAST case scenario if all other avenues (after months or even a year or two of doing any one of the above) have exhausted themselves, look into an autism assessment. Many people who find themselves in difficult mindsets are simply neurodivergent and knowing this can bring a lot of peace, and a more open minded look at how to tackle the issues it presents. To a small degree, you can even begin to bring in some tools that might give you hints. When angry, and frustrated try giving your partner (and perhaps the child) a small squeezy ball (dollar store!) to squish. See if that feels physically calming, or stimulating, or mind clearing... It's going to take a few minutes to notice any possible effects and probably will take some time for the persons to discern what if anything it may be offering them.
Lastly, ELIMINATING stress is not allowing the person to learn to deal with stress. It's allowing the person to think that avoiding stress is the only option for emotional regulation. This is something the two of you should talk about. Yes, when there is stress you get symptoms which can be anxiety, anger, or physical sensations. You deal with the inner stress (it's not just about childhood, it's about how these things make you feel about yourself now, in present day) by utilizing tools you've learned work for you, and then the symptoms begin to fade. This is normal, it's just the interpersonal tools each person uses vary.
Again, please encourage your partner to join us and ask questions. They need a private space to be able to express their own desire for assistance. It's fantastic you are so supportive and a wonderful advocate, but it's really up to your partner to deal with this stuff themselves. Dr. David Hanscom a TMS practitioner discusses in detail how important this aspect is. It's about changing the brain's values from possible perceived victimhood (again, the conscious mind might not even know this is happening) to having power over your own brain and mind.
The challenges are actually fairly simply to work with, it's just finding the correct tool to make a path forward that is needed.