how to go about this in terms of accuracy would probably go:
1. determine if there was a region from which "the pilgrims" (how many groups of pilgrims departed for the new world at different times?) originated. looking very quickly, it seems like the separatist religious groups they were part of were from nottinghamshire (so nottingham yeast? lol)
2. research brewing in england in the early 1600s (afaik from that time it was really just "malt" with mixes of malted oats, malted wheat, barley etc. i dont know if they bothered roastng them. but ?? . hops were likely used in some beers in england, i think at that time there was still a fair amount of unhoppt beer.
3. research the situation of the pilgrims in america immediately after landing and 5, 10 years after landing. would they have had corn crops going by 5 years in? would they have attempted to use the corn for brewing? did they import barley from britain or did they grow it? could they afford to use these crops for brewing alcohol or was their food situation too tenuous? did they choose to not imbibe alcohol or only take a little bit?
4. did they drink beer at all. apparently for example beer only became popular in ireland relatively recently, so the default alcoholic drink for centuries before was in fact whiskey. i know in early america (but later than pilgrim times) apples were used to make cider and apple brandy, rum was very popular etc.
i think considering the implications of their life and understanding it would be better than making a beer that they would have had. im guessing if the pilgrims drank beer (which i believe they did) it would have been a quite boring and earnest product, likely just some malted barley, perhaps not even kilned, just air-dried. maybe mixed with corn grits, some other grain or a sugar source. hopped? maybe? i have a feeling the gravity would be pretty modest. yeast? probably the same they used for bread making.