What to do about a no win situation of performance punishment?

A bit of background, I am a high performing software developer working for an organization that provides unbeatable benefits. I don't plan to leave as I am happy with my compensation package and the overall culture of the organization.

However I seem to be in an impossible situation where I'm one of few employees on my team who is capable of delivering on difficult projects. HR at the org has created a compensation hierarchy where there is only a management track beyond my level, and no real means for me to be compensated for the value I provide. If they move me into more of a tech lead / management position they have no one to execute on hard projects, if they leave me where I am I'm doing difficult work that some of those who are senior to me (making more money) are incapable of. Some of those senior to me were promoted because that was where they could be useful: attending meetings and not doing much technical work.

At this point I am highly respected across my department due to my ability to deliver, but I feel increasingly frustrated that I'm a magnet for some of the hardest work coming into my team while not being recognized monetarily.

Many months ago I approached my manager and explained to them that I am being over-relied on and am approaching burn out, but given the composition of my team there isn't much they can do about it, I'm their only option for many projects.

The question is: is there any reasonable approach I can take to improve this situation? It feels like I'm stuck and I'm not sure what can be done about it.

Edit: I do realize this question might not be a perfect fit for the Stack format, but at this point I just need anyone to run this problem by. It's been bothering me for a long time.

Corporate guidelines are guidelines. They are not requirements that must be followed or the world turns into some "Mad Max" like post-apocalyptic and dystopian..... (enough histrionics)

The point is that they (HR, Management, etc.) can create a new title. It might not be quick or without going through levels bureaucracy and politics.

What you are describing is being caught between career growth and a lack of a defined path at your current company. Making the point that in a field that is changing as quickly as dev and with someone like you that has domain knowledge, that the current "guidelines" place an artificial ceiling on your growth.

Making the case that going the management path to you will give you recognition of part of your skillset, but also cost the company of the value of the part of your skillset that is most unique.

Consider writing up a job definition that you are suggesting to them. Giving them a job definition and the case why you should be promoted into it (all you wins, etc.) demonstrates that you want to stay with the company. It is much better than walking into your manager's office or HR and saying "I want a raise or I'll leave".