Quite a few of the so-called 'natural' or 'organic' permanent box dyes (health food store type brands, still loaded with artificial chemicals tho) are PPD free; some of the Garnier Nutrisse lines are which is what I use. I would be surprised if Sallys didn't sell something. Anything that claims to cover grey is highly likely to contain PPD.
Not sure if that is a typo but no there is no certainty that eventually you WILL become allergic, you jut might and it can happen at any time as with any other allergen
of which there are many in skincare and haircare products. Severe reactions are very rare, not saying you should not be cautious you absolutely are right to consider it. But most people who have severe reactions didn't bother with a patch test and/ or previously had a mild reaction which they ignored.
Your technique would take forever, so some hair would be processing for a short time and some for a long time, you risk a lot of damage which could cause you major problems with porosity and breakage over time. That's if you could get the stylist to agree to it.
Do you really need to permanently dye your lengths each time? Could you permanent the roots then semi permanent the rest? I appreciate that doesn't resolve the problem entirely but it does shorten the working time and there would be less goop on your head to wash off. Your colourist might also put a barrier product on your hairline if you ask.
If you are not already start avoiding sulphates and other harsh anionic surfactants, not just in shampoo but in face and neck products. Sulphates are proven to thin and dehydrate the skin at concentrations as low as 1% which can leave it more vulnerable to irritation and allergies. You can also reduce the risk of future allergies with diet modification: think anti inflammatory nutrient dense wholefoods.