If you're lender asks you to submit proof of your current income then you need to do that to complete your PAYE application (by the way, have you looked into REPAYE? The interest deduction is much better).
I don't know of many people that were able to get the $0 payments for their first year of repayment--people were able to get away with that when IBR/PAYE first came out, but now loan servicers generally ask for your current income as they know people are working. I don't know why your school told you that your $0 M4 income would be used--it's certainly possible that a loan servicer might accept that, but it's rare. That was bad advice for them (and the consultants) to give you. Besides, you also have to answers a yes/no question on whether your prior year's tax return accurately reflects your current income--you really shouldn't be answering "yes" if you have an income now.
If you were already approved for PAYE and your monthly payment is $0 then you may be ok. But maybe the representative you spoke with on the phone documented you need to submit proof of your current income--maybe that impacts things when you submit your tracking form, maybe it doesn't.
When I applied for IBR I sent in a paystub from my intern year so that IBR was based on my current income. I figured, why risk it? If PSLF sticks around, I want to benefit from it as soon as possible. Saving a few hundred dollars per month my intern year just wasn't worth the risk of not getting PSLF. Besides, I was going to have to make those full payments the following year when I reapplied for IBR, so I figured it'd be better to just start intern year so I don't feel like I have to ramp down my lifestyle when the larger IBR payments started--it's much easier to just maintain the same lifestyle than have to ramp it down $300-$500/month.
I second the idea that you should take a look at REPAYE vs PAYE as long as you don't have a ton of accrued interest that would capitalize if you switched. REPAYE is a great hedge against the elimination of PSLF