...My insurance company is a big company you would've heard of, the other company is tiny and I've never heard of them before now. From my Googling it isn't clear to me if my insurance company will be able to (or want to) escalate once I am out my deductible....
I have been in your position and I filed the claim with my own company and paid the deductible.
To address your question above. I'm going to call your company A and the other company B. Company A got money from you (premium) and they paid a covered loss. Right now, they have a cash inflow of say $200 and a cash outflow of $1,500 + expenses. Company B collected premium from the other driver. The other driver has now had a covered loss. They have collected the same $200 from their driver, and they have a potential liability from the crash of $2,500 + expenses. Right now they have paid nothing.
If you are company B, you don't want to pay company A, but there's a problem. Company A is likely to be in an opposite situation with Company B on more cases. If Company B doesn't pay up on the subrogation in your case, company A might not pay up on the others. This is why your company is in a much better position to get the deductible back than you are.
If legal liability can be attributed to the driver of company B, I expect that they will likely get a decent chunk of your deductible back. In my case, I got it all back, but it took 6 months.
I can't tell you what to do exactly, but I would expect your company to be in a better position to get it back than you. Your alternative is a lawyer, but over such small amounts, you may have difficulty finding one where the lawyer comp doesn't destroy the value of the recovery. Just my $.02.