I'm sorry to hear about the lack of progress. Honestly, you may need to accept that court will be the only option.
I would not start your application seeking primary residency. Just start out by requesting reasonable time with the child and then wait and see how the mother handles it. It's important to give the perception that you want to co-parent with the other party, even if they refuse to do so. If she ignores court orders or tries to frustrate the time the child spends with your partner, then you might have a viable case for residency. The court has become surprisingly obliging of the 'friendly parent' doctrine, quite often ordering residency to the parent most capable of supporting the child's relationship with both parties. However, you need to prove the mother is not the friendly parent over a long period of time, and the best way to do this is to observe how the interim orders are played out.
If the current proposal has been pending for two months with no response, then it's clear she has no intention of responding and it's time to take the next step, which is an initiating application. Filing an initiating application is a big step for you, but it's often a very intimidating step for the other parent, too. They will undoubtedly get legal advice, and 99% of lawyers are very honest about how family law proceedings go - it's unpredictable, withholding is not good for a case, and the child will likely be granted time with the other parent.
In terms of money, I would argue you won't need to engage a solicitor unless it reaches final hearing. Interim hearings are very easy to navigate as a self-represented litigant, and it seems to me that you have a reasonably good grasp of the legislation, so I would hope your partner has enjoyed the benefit of your research, as well. As an FYI, my partner self-represented on my knowledge of family law for three interim hearings, then the other party sought settlement two weeks before the final hearing date. The consent orders very largely reflected what my husband was seeking if it were heard at trial.
As you have highlighted, you also are not the only party looking at a money struggle. In our case, the other parent had already spent $10,000 on a lawyer, an additional $4,000 trying to negotiate consent orders in the two weeks before the hearing date, and was told it would cost an additional $8000 if they went to trial. That's an enormous amount of money for a single parent working part-time. In contrast, our total expense, from beginning to end, was $2000, so it was the other parent (well, her solicitor, actually) pushing to settle in the end, even on things she was unhappy about. Self-representing can be challenging, but if you're willing to put in the time and effort to articulate your case, it can be very worthwhile, especially if the other party has been unreasonable such that their case is weak.
Forgive me for forgetting, but have you already attempted family dispute resolution?