FILE THE CLAIM. Forget getting any sort of verification or agreement or information from the employer. Let the unemployment system do this. After being FIRED from a full time job, you are in NO WAY obligated to continue the work relationship with them or lose benefits. It is not considered suitable work that you have refused if your employer fires you, and then says, "Oh, by the way, if you want to you can come back and work for us on call." As it has been pointed out, they know that if they do that, and you do accept the on call occasional work, based on the way NY figures it, it will greatly reduce your benefits.
I repeat, always, let the unemployment system make all these decisions, don't try to have everything figured out before you begin the claims process. You do not have to go in and tell them how to do their jobs. And many many times, if you delay filing to obtain the predictions of people on the internet, you have messed up by not filing quickly, or by working at the undesirable low ball part time thing they have offered you, even for a minute. Or like someone here recently, you develop a misconception of how the claim must be taken and end up telling them to just keep their lousy benefits, etc.
But for sure, I tell you, DO NOT accept and work at this part time on call thing. This is just the employer trying to evade the whole unemployment process. If you were fired for a valid misconduct reason related to your work performance or a circumstance, then they should not want you back part time and on call either. Regardless of the prevailing wage and all those other factors, their offering you this employment part time after firing you for a supposed valid misconduct reason is not a valid job offer of suitable work. It is nothing but a shady way to try to stop you from drawing at all.
Of course they will "fight" the claim. That does not mean they will prevail, or that you shouldn't file the claim. It just means that they will be out some money if you do draw benefits, so no matter what the circumstances, the company doesn't want you to draw and will usually try all sorts of evasive tactics. The unemployment system has seen it all before.

