Perhaps I can help.
Except for your first bullet under target qualifications, there is NOT likely to be a database that shows a firm a) moving to more complex projects, b) looking to restructure services, c) taking on new or unusual projects, and d) struggling to find a process that fits their needs.
What might be possible is an annual list of dev/design agencies with their contact information and their SIZE. Then, if you had several years in a row, you could see which ones were growing.
But until you figure out if that listing exists and whether you can get one, you need to do the hard work of finding ONE such target company, review what you did to find it, figure out if that is repeatable, if so, repeat it, and if not, modify how you found the first one and find a second one. Rinse and repeat. If you already have a few prospects, then you are there and you can ignore the next paragraph or two.
If you do not have the first prospect...
The first question that jumps to my mind is...how about all those agencies that you communicated with to determine that you needed to call on agencies with the five characteristics you posted?
You suggest you know fairly precisely who can benefit from your product or service...if you did this by talking to agencies, then call on those first...some would be prospects, right? Or were they telling you what other agencies would want, but they don't want it? Those agencies could at least tell you who they were thinking of when they told you what was needed.
I fear -- based on your question -- that you have determined your target without actually communicating with any target prospects. You may very well be a fantastic guesser on their operations, problems, and the features needed to address them, but it's not likely.
To summarize so far:
If you have validated your offering with prospective customers, then start with them. Also, leverage their knowledge of the industry to figure out how to find others. That would work with agencies outside your target market, too....they may know firms that fit your profile.
Figure out what organizations they attend, etc. Who they would call if they needed extra capacity, etc.
If you do not have those connections, then your target characteristics for your product are somewhat suspect and you thus you will NOT likely be wasting time to call on ANY local development/design agencies.
Local, because then you can visit them in person. After two or three meetings, exploring with them whether they have the problem you solve (they might), you can also explore with them how you might find others that meet your criteria.
You just have to do this hard work, one company at a time until you discover a pattern or a source. The fastest way to get to a lot of companies is to find the first one...not try to find a list of 50.
Another way is to talk to a senior person at a company that also sells to your target market and ask them how they find them. This is also a question when you meet with the agencies...what else do you buy, who do you buy from.
Contact me directly if you want clarification or want to discuss in more detail.
Can you see this would NOT be a major issue if you started with solving a problem you know someone had and then did customer discovery...because you would be developing your sales funnel as you developed the product?
However, if you are at stage 0 prospect-wise, then you need to realize there is no short-cut to finding the bull's-eye. First you find one, then two, then you figure out how to find two-at-a-time, then more, etc.