Having found no information online I asked Elio and Linda on twitter and they've come back saying it's a case where the knight and the "knighter" probably weren't that bothered about having to follow the Seven.
@CarrotLethal: Hello! How is Ser Bartimus a knight if he keeps to the Old Gods? I know the Manderly's keep to the Seven but isn't it the knight that needs to keep to the Seven? Thanks.
@westerosorg: Good question. He seems to be an example of someone who was offered knighthood as a reward for his services, was willing to say the words for the value that the honorific gave, but was not really a believer at all. Whoever knighted him did not seem overly picky on the matter.
@CarrotLethal: So it would appear that in modern day Westeros being a believer in the Seven isn't really necessary to become a knight? Either that or this is an edge case?
@westerosorg: More an anomaly rather than normal. But given that we know from "The Sworn Sword" of unscrupulous knights who'll knight other men who bribe them, it seems like there may be reason to think that there's certainly room for non-Seven following knights here and there.
This can be seen as we see two forms of the oath the formal version which is more religious.
The Laughing Storm gave an impatient shake of the head. "Go to him, Ser Duncan. I'll give squire Raymun his knighthood." He slid his sword out of his sheath and shouldered Dunk aside. "Raymun of House Fossoway," he began solemnly, touching the blade to the squire's right shoulder, "in the name of the Warrior I charge you to be brave." The sword moved from his right shoulder to his left. "In the name of the Father I charge you to be just." Back to the right. "In the name of the Mother I charge you to defend the young and innocent." The left. "In the name of the Maid I charge you to protect all women."
The Hedge Knight
Then we have a less formal version which doesn't really seem to touch on religion at all.
This time the lightning lord did not set the blade afire, but merely laid it light on Gendry's shoulder. "Gendry, do you swear before the eyes of gods and men to defend those who cannot defend themselves, to protect all women and children, to obey your captains, your liege lord, and your king, to fight bravely when needed and do such other tasks as are laid upon you, however hard or humble or dangerous they may be?"
A Storm of Swords, Arya VII
So it's clear some knights and lords are less restrictive about knights having to be followers of the Seven. We also have the case of some more knights in the North but it's unclear if the follow the Old or new Gods personally.
- Ser Jorah Mormont: Although House Mormont keeps to the Old Gods it's unclear whether Jorah does personally.
- Ser Rodrick Cassel: Like Jorah it's unclear what God's Rodrick follows.