@Matt Sully Apologies for all the inconvenience caused!!!
Assuming from your query that you are talking about Azure Support you can correct me if I am wrong.
Now there are different kind of Support Plans in place you can make use of. More info here.
Considering the scenario you mentioned where you are not getting the required help and you want to escalate it. Now when it comes to basic tier as mentioned by @Cheong00 above you can ask the Engineer to escalate internally if you want it. They do have that internal system to escalate which should work for you.
Another tier is the Premier one which works on same mechanism. Customer with Premier account have Account Managers as well which can help you to escalate the whole process to get the required help.
There is no external process as such where you can escalate but internally it is possible. I will reach out to you separately and will try to help you to get it escalated internally.
Hope it helps!!!
Please "Accept as Answer" if it helped so it can help others in community looking for help on similar topics.