I remember your situation very clearly. I am really sorry that your son is continuing to struggle with his speech & behavior. I can appreciate how HARD that must be for both him and you.
I can just hear the anguish in your post.
Please forgive me for the 20 questions, but I'm asking so that maybe I/others reading this can give you better advice.
I remember that you were able to get a limited amount of time with a very senior speech therapist at the Lizard Centre (who works almost exclusively with kids on the spectrum). Did she have sufficient time to do a full assessment of your son's speech? And did you have time to brief her on the other concerns you have (beyond lack of speech - the sensory issues, the tantruming, etc.), or was the focus solely on his speech? What were her explicit recommendations to you based on the time she spent with your son?
I can't recall which paed you saw, but did he/she do a formal assessment for ASD (this would have included at the very least a lengthy parent interview and a play based observation session of your son)? Was it a developmental paed or a generalist paed?
I am asking all of these questions not because I am trying to grill you but because sometimes parents think things like ASD and other developmental issues have been ruled out completely when they actually haven't been fully probed. If you did have a full assessment, then please just disregard the line of questioning above! And if you didn't have a proper assessment, I would strongly encourage you to seek the counsel of a good developmental paed (specialist paed) as soon as possible. Since your appointments were several months ago, and your son apparently has had a regression in skills, it is important that professionals be aware of that.
The "diagnosis" that your son is "willful and spirited" (and that is the root cause of his behavior) just seems -- well, lacking, frankly. From a good paed, I would have expected some concrete suggestions on how you might address the behavioral issues and other potential issues to explore, even if the paed didn't believe ASD was in the cards. Likewise, I would have expected L to have given you some recommendations with speech strategies, even if she didn't have capacity to take your son on as a client.
bubba raises a great point about the "fit" of your current speech therapist with your son. It may well be that he needs a different therapist or much more therapy...or both. And like bubba, we saw enormous behavioral changes with our daughter once her speech started to kick in and so did the effects of ABA (behavioral therapy).
I wish I had a magic answer for you. I think if I were in your shoes, and depending on the answers to the questions above, I would simulataneously be:
* Booking a great developmental paed (again, assuming the paed you saw was not a developmental paed and/or didn't do a formal assessment)
* Looking for a speech therapist who tends to work with kids with ASD (not because your son necessarily *has* ASD but because these types of speech therapists will be totally familiar with things like severe receptive & expressive language delays, extreme meltdowns, sensory issues, running off, etc.)
* Trying to get more support in the home -- be it in the form of more help from my partner (edited to add that UKBlondie is spot on!), family members, or at home therapists to help lighten the load.
Good luck. I really hope that you start to see the sun from behind the clouds soon. Three was a very hard age for us on this end, too. But with a lot of early intervention, we did see great progress. I hope things improve soon.