The source of the powers of an oathbreaker

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):

1. 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.

2. 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?

3. 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!"

   or

   "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.

4. 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.

The holy avenger was officially part of the Forgotten Realms since the beginning.

The official beginning of the Realms in a game publishing sense¹ starts in 1987 with the Forgotten Realms Campaign Set—the “Grey Box”—published for first edition AD&D. By that point the holy avenger was already an integral part of AD&D, appearing in the DMG (1979) on page 165.

Note that Supplement 1: Greyhawk is a 1975 supplement for the earlier, original D&D game (dubbed “original D&D” or “0e” (zero-e, oh-e)) published in a box of three stapled booklets in 1974. The holy avenger by any name was a well established part of the game before AD&D or the Forgotten Realms were even conceived of as publishing products at TSR.

The 1987 edition of the Realms predates the idea of AD&D edition changes and the accompanying cataclysms in the Realms. Those were later inventions by minds at TSR after Gygax had been forced out. This makes the holy avenger part of the baseline Realms from which every edition change and cataclysm later derived.

So for the purpose of tracing the origin of the type of magic sword, it’s integral to the Realms, and its AD&D description is the baseline version, before any Realms events that changed how magic works.

1. A note on when official-ness applies to the Forgotten Realms. The origin of the Forgotten Realms predates D&D entirely, circa 1967, as it was an invention of Ed Greenwood’s childhood imagination before he encountered D&D. It’s not possible that anything derived directly from the D&D holy avenger was part of this pre-D&D conception of the Realms. Even when Greenwood was publishing his home game’s details in Dragon articles, it wasn’t “official” at that point even if his home game likely had a holy avenger at least in their rulebooks, as Greenwood’s articles were just a fan sharing his ideas and Dragon at that time often published fan/homebrew material that wasn’t endorsed as “official”. As Greenwood wrote more articles, the line between official and fan eventually blurred, to the point that TSR bought the Realms and published a novel and the Campaign Set in 1987. This answer uses that year as the most meaningful and clear beginning to the “official” Realms. The very first official publication was actually the novel Darkwalker on Moonshae, but only by a month since it was written in tandem with the official campaign set, so that doesn't impact the 1987 start-date.