My question: Is this possible?
Yes, it's possible, whether or not the fiend and the god/goddess get along. There is no RAW prohibition from the multiclass, no matter how awkward it looks.
From a purely RAW standpoint, I don't think that Warlocks can lose their powers, but would his good deity even bother with someone who sold their soul to a devil?
This depends on how the DM plays out the deity's and the fiend's role. Points to raise as you two flesh this out (a collaborative effort between you and the DM):
- How much of a role does redemption play, a deity accepting a lost/troubled soul (bound to a devil) into the fold? The Sword Coast Adevnturer's guide supplement for D&D 5e has some material on FR deities; Baervan gets one sentence of treatment (p. 115) and Chauntea gets a few paragraphs( p. 27). For more detail, you could back up this guidance for Chauntea or Baervan with previously published material and make your case to the DM for why, or why not, either deity fits. Deities thrive on having followers.
Playing an internally conflicted character is rich with role playing possibility. Trying to serve the good, still dealing with evil ... that's a standard story IRL, and an element of stories from many different cultures. This would allow the link with the Pit Fiend to remain -- but there's always a price to be paid! (This could provide DM some fun, occasionally at your expense! :) )
And would that pit fiend continue fueling their powers?
Why not? Here's a point of view that the Pit Fiend could take:
"I've got my hooks into this gnome, I won't let him go. So he wants to get in with that goddess/god? Good! I can use him to cause trouble for (chosen deity), and then enjoy his suffering as he realizes that it's his fault -- due to his hunger for the power only I can give him. Muahahahahaahaa!"
"I don't like (deity), but we both have (unrelated) unfinished business with that #@!^%$ Orcus, and this gnome will serve nicely as a proxy. In the end, I profit! Muahahahaahaa!"
You and the DM should be able to work out a deal, and some tension, that fits the campaign.
One last point to address as you work with your DM on this multiclass:
How important is alignment in this campaign?
Alignment in 5e is a bit different than in previous editions. It is more like an ideal for the character, and a matter of how the player should behave. How well the player character lives up to it, and what the rewards/penalties are, lay in the realm of DM's discretion.
With that in mind, and the two deities to choose from: Chauntea's clerics appear to require a serious commitment, and a decision on being a Pastoral or a True Shaper (the latter looks like a better fit for an adventurer), while Baervan's yoke looks like it is lighter to bear for an adventuring cleric.
Could the pit fiend and the god/goddess get along?
That would require a serious threat that both of them want to counter, something opposed to them both for different reasons ... so that you would be serving both of their interests (roughly) at the same time. Some existential threat to the world itself ... that too lies in the realm of how your DM is running and shaping the campaign.