I have been at this crossroad before. Client likes my work and requests something my equipment can't handle. I upgrade and continue until the next request comes along I can't meet because of some limitation. I've done it twice.
There is the old proverb, "don't put the cart before the horse."
I have been of the mindset to let the horse pull me along until I need a bigger cart. But only after I've exhausted stacking the old cart as high as it'll go.
In all situations it is always about problem solving and finding solutions that don't lead to other problems. So if I understand your problem correctly you have come to a point where you can not get close enough to your subject (lens restriction) to fill the frame (4/3rd frame) to give you a large enough image size (pixel loss) to print large enough (larger print size demand).
Possible solutions are the obvious; get a longer lens or a camera with higher resolution. Both require a significant monetary outlay and, in the case of the higher resolution, may not equate to higher image quality anyway. That can lead to additional problems as it puts a financial strain, etc.
Have you tried less expensive solutions? For example, getting closer to your subject? If your bird images are what's in demand from you and you need to fill the frame more with those subjects perhaps you going to them or them coming to you is a less expensive solution. Taking a lesson from nature videographers, they will stage a scene and use every trick in the book to attract their subject into the frame. This includes using blinds, remote camera operation, baiting and creating attractive 'sets' for them to perch on. Of course you need to put in your due diligence to find out each species' habits in order to predict where you need to be and what attracts them in order to get the desired shots, but this is part of what being a pro photographer is all about.
One thing for sure, you are on the right track in talking to a pro printer to determine what technical requirements are needed for larger prints. It's not all about pixels as there are other elements at play; viewing distance, printing substrate, print technology, etc. For example, I did a job where one of my images was to be printed 6 feet tall and however many feet wide. I used my Canon 7D (crop sensor) for that shot and the image came out great.
In this case I photographed this gym's Zumba instructor on location just prior to one of her classes. It was shot horizontally and a lot looser than what was used. As you can see in the image above she was cut out of the image, cropped tighter and dropped onto the purple and red 'wave' background. Understanding the end use goes a long way to knowing what you need to capture.
Hope this helps a bit.
P.S. I hate this shot as it was done without any art director or layout. All I knew was that it was to be printed large on a sun shade.