Leaving Epson pigment ink (or any pigment ink) in a printer for a year is going to produce issues from inside the printhead backwards - through the ink exchanger and its damper filters, through the ink lines, and even possibly back to the port stems that pierce the remote ink tanks. So therefore, cleaning under the print head will not really do anything. That's best as a preventative maintenance routine you should do every 3 or 6 months along with the clean station. If again you find 3 months later that you have not printed - then install PiezoFlush into the printer and then you can be a bit worry free about when you will start it up again.
I would not try and clear it using more ink. Inks are not solvent in the way that PiezoFlush is. So, I do not believe that you should waste the ConeColor ink to try and use it as a flushing agent.
I believe your best opportunity to save this printer is to install PiezoFlush into the CISS and to run an INITIAL FILL (twice off the start) to remove out the thickened OEM inks. The first will bring PiezoFlush to the head but it may also (because of the thickened OEM inks) not travel as it would if unimpeded. The second will allow it to travel through the system without being impeded by the thickened OEM inks. Each INITIAL FILL takes about 8 minutes to complete.
Then turn off the printer for 48 hours to let it sit with the PiezoFlush inside. Then run the INITIAL FILL a third time followed by your first nozzle check. Hopefully you will have cleared the printer.
But, I would still have a worry about the black ink exchanger (matte black / photo black) and its inactivity. If the printer appears to have been cleared you would then need to operate black ink changes until both mk and pk show all pink nozzles. The INIT FILL does not clear the parts in the black ink exchanger that prevent MK and PK from intermixing. Both of those inks share one of the channels in the print head. The black ink exchanger does the switching. So you can see the problem with allowing that part to sit for a year.
If you are not up for something like this - you can probably get a repair center to replace the entire ink exchanger. It's less than $200 part and you would pay their labor. As long as the ink carts are not expired they would only need to replace the low ink carts to test the print head, etc. So consider that cost at the center. They have a fluid similar to PiezoFlush but are more inclined to replace print heads rather than remove them and use cleaning fluid inside of them... So the potential is for an expensive repair that includes ink exchanger and print head unless you find a center that tries to save you money.
In any event - there may not be an easy fix. But, if all of this sounds like a hassle and you may bin the printer - you have nothing to lose to try and clear out the old Epson ink with ConeColor. You can always transfer the CISS to a new P600. Run 2-3 INITIAL FILLs. It uses powerful suction and does not over stimulate the print heads. You just never know....
Well there is your morning food for thought. Do the solution which to you seems most feasible.