OED puts the origin of "putting" to describe a golf club before "putt" or "putter."
1690: This is not that I doubt but ye made good use of your short putting club ther.
1776: Nor must any person whatever stand at the hole to point it out or to do any other thing to assist you in putting.
If "putting club" came before "putt," It seems possible that the spelling of "putt" with regard to golf was irrelevant until its use as a verb began, which would make this use distinct from "putt" as attributed to c. 1300.
Notably, OED cites the origin of "putt" as a variant of "put."
Origin: A variant or alteration of another lexical item. Etymon: put v.
OED Public also has an article on Middle English indicating that such doubled consonants often were used after a long vowel along with a silent 'e.' (Although, of course, the point about long vowels runs contrary to how we use doubled consonants in modern English. This point may be specific to the use of a silent 'e,' like with "butte" in modern English.)
iii). The final ‘silent’ –e was much more commonly found, not only as a marker of a ‘long’ vowel in the preceding syllable (as in take), but with no phonetic function, and sometimes after an unnecessarily doubled final consonant.
Also it is to be noted that this crosse made & gyuen vnto the newe crysten man is the seuenth crosse & the laste that is sette on his body.
It cites the etymology of "put" as deriving from Old English.
*The existence of a verb *pūtian or *putian is implied by the derivative Old English pūtung or putung (see putting n.1). Regular phonological development of Old English *pūtian would give Middle English pūten ; compare the forms at Forms 2. Old English *putian would probably have given in northern dialects Middle English *pōten , with close ō , as a result of Open Syllable Lengthening; in southern dialects, poten could also have occurred as a spelling for puten , especially in later Middle English. However, such forms would be indistinguishable from the reflex of pote v., and all such forms have been placed at that entry. The modern form put with short vowel probably results from generalization of the short vowel of the past tense and past participle. (If the original form was pūtian , the past tense and past participle would either show trisyllabic shortening or shortening before the doubled consonant in syncopated forms.)