Hi Mark and welcome to ForestConnect! I don't have the answers to the technical questions, but here is a resource on MyWoodlot.com for a video on hack and squirt for Tree of Heaven. https://mywoodlot.com/index.php?option=com_zoo&task=item&it...
Thanks for the question.
The NYS DEC maintains a website that lists all of the products (herbicides) that are legal for use on specific plants. I find the site quite useful as you can search for a pest, such as Ailanthus - tree of heaven, or by chemical or by EPA registration number. I've pasted an image below of the search I did for your question.
In my search I found basically two products that are labelled for use in NY to control Ailanthus. Both have the active ingredient triclopyr, which for both is usually mixed with an oil base. Pathfinder II is premixed. The other is Brushtox and appears to be a concentrate that you will need to mix. I attached the labels for both products to this message. I have used Pathfinder II, but I am not familiar with Brushtox...maybe another reader is and can comment.
I believe you can purchase Pathfinder II from arborchem.com, and at a discount if you are a member of New York Forest Owners Association. You could also search the internet for local vendors for these products.
Here is the web site to look for the labels. NYS DEC pesticide (herbicide) labels
Note that neither of these products is labeled for use via a hack and squirt treatment. Both can be used as a basal bark treatment or cut stump treatment. In NY the pest plant and the manner of treatment (e.g., hack and squirt) must be listed on the label for the treatment to be legal. There are other resources on this site for basal bark, if you need more information. Search for basal bark.
Let us know if you have more questions.
In addition to all the guidelines provided, it is important to note that this is a difficult tree to control. Penn State has prepared a very helpful fact sheet on control of Ailanthus which you can find here Remember that you'll need to use the pesticides allowed by NYS DEC and follow all the label instructions.