Will I have roadblocks if I'm filing for POA too late?

You don't file for POA. POA is conferred on you by the elder. That is to say the elder asks you if you will serve as POA, or you discuss it with the elder and come to the conclusion together that things are being missed, and you together decide to see a Lawyer to have a POA done. This is done by a rational individual who is NOT diagnosed with dementia, or who is facing early diagnosis but currently rational and understands what he is doing by a POA. So this should have been done when you knew you might need it.
It wasn't, so this is where you are now.
You should see an attorney. This is not a case where you can pull papers off a computer and do this on your own. This attorney will "examine" your Dad to see if he understands. My brother decided to make me his POA and the Trustee of his trust when he was diagnosed with probable early Lewy's Dementia, but still in full control of his life, and knew what could be coming. We went together to his attorney and the attorney examined him ALONE, and with me in room saying "Do you trust your sister implicitly because you are giving her the power to sell the gold out of your teeth if she wishes to".
If your father cannot fully understand that he is giving over all financial and placement power to you, then you would need to forget about a POA; it is too late. You would need guardianship. If your father fought guardianship he would win (especially if he is "working".)
You are in a bad place. Step one is now to discuss with Dad. Does he want you to have full bill paying powers and to be on his accounts as his POA. If so, do YOU understand you are a legal fiduciary responsible for keeping meticulous files and records and are you willing to do all the work, all the mailings, all the filings, all the phone calls to all the entities? If so, go to a Lawyer together.
Do not go for an official diagnosis until POA is done, if it CAN be done.
I wish you good luck going forward.

I think you first need to get Dad to a doctor. He could have a UTI or something physical. Low potassium, thyroid, dehydration, diabetes can all cause Dementia like symptoms. An MRI will show any Dementia. If Dementia is found, then ask the Dr. If he thinks Dad understands what it means for him to assign you POA. If he says yes, then get to a lawyer right away. Decline can happen overnight.

If you can get POA, ask that it be immediate. That means as soon as Dad signs the paperwork, you have some control. Remember though, you are Dads representative and as such your decisions should be based on how Dad would handle it. Its a tool not so you can control him. The other type of POA is Springing which means its only in effect with a formal diagnosis from a Dr. POAs of this kind are always Durable. This means for life unless revolked by the person who assigned you. POA is revolked at time of death. You want financial and Medical. Pick a time of day he seems more lucid. Late afternoon and evening hrs they usually "sundown" making things more difficult.

Judges have nothing to do with DPOAs. That's between the person assigning DPOA and their lawyer. Guardianship goes before a judge and his ruling is determined by the doctors who have examined the person and shown that tests prove the presence of a Dementia and the person is now incompetent to handle their own affairs.

Diskidmore, no one can "file" for any kind of POA or DPOA. It's only GRANTED by the individual who wants someone to act on his/her behalf.   This individual also sets the parameters for the proxy (which you would be if appointed by your father) and authority which he's willing to grant to you.

In the meantime, you don't need to be a proxy to straighten out bills.  Gather them, start charting or logging receipt, paid dates and amounts.  If he's paying with a checking account, you can write the checks and have him sign them.

But if he's cognizant enough, you can also go to his bank with him and get new signature cards issued with both your names on the checking account.   This has good and potentially bad repercussions, the bad being if his spending isn't rational, or if he's in debt.  

It might help to do some background research on powers of attorney so you understand what the parameters really are.

"Still working" is apparently only technical at this point. He hasn't been fired. His boss is a good friend and Dad was a good employee for many years. He pulled dad out of retirement multiple times. Dad's brilliant, and uniquely qualified, if this is recoverable he's worth having a contract in tact with.

I should bring forward his doctor's appointment if I were you - when is it?

Whatever's wrong might be an acute infection and once he's recovered he'll be mentally fine and able to create a POA with no issues. But at the moment, if he can't concentrate for long enough to deal with one little bill, it really doesn't sound as if he has the mental capacity to understand what he is doing in order to create a valid power of attorney.

I had a discussion with an elder law attorney about this. People are only incompetent (legally) if they've been declared incompetent by a judge. Doctor's opinions count, of course, but the judge decides. So basically your Dad is competent until proven otherwise. You said he's still working, right? Make sure it's a durable POA, without the ability to be assigned to someone else ahead of you (like a nursing home). Try to get an elder law attorney. It's really worth paying for their advice.