Question - 

Does Tiamat have any rival deities in any pantheon who are specifically Evil ? It sometimes feels like "all" evil dragons worship her, and that the only gods who are brave enough to actively oppose her are Good. 

Does she have any Starscream-to-Megatron types? 

There is an entire draconian pantheon to draw from, though the other members aren't so well explored.

You might be interested in:

Task - God of Greed

Garyx - God of Destructive Power

Null - God of Death and Undeath

5e doesn't have much, if any, support for them, so you'll need to homebrew whatever you use.

For a "not-quite-god", you may like Daurogoth, who is a dracolich chosen by Mystra as a necessary evil for magical innovation.

You should also consider archdevils like the princes of hell as her rivals. While she isn't really part of devil hierarchy, her influence in hell is comparable to theirs.

Nearly all evil dragons at least respect her as the progenitor of their race and incredibly strong. Daurogoth would be strong enough to be a threat to her plans (if he chose), but not to her existence. Even an immortal chosen isn't as strong as a god (even a lesser one).

Multiple editions have had lore indicating that there was once a rival to Tiamat whom she slew long ago, but in the present she has none.

Dragons tend not to be terribly religious in the first place, so there isn't much need for more evil dragon gods.

Gotcha. 

Basically, i have two factions of NPCs who will be in conflict. (Technically they are Empires.)

One of the factions strongly features a cult of Tiamat worshippers.

The second faction is also "evil", and i thought it would be cool if the reason for this conflict was undermining Tiamat. So if they were worshipping a deity who was already going after her, that would be ideal.

Evil dragons tend to be both arrogant and ambitious. I could see a powerful dragon scheming to usurp Tiamat's title.

I played in a campaign where Tiamat had defeated Bahamut and we were trying to resurrect him. At one point, we uncovered a plot by a powerful Mithril Dragon to intervene at the moment of resurrection and take Bahamut's power for itself. It reasoned that if Bahamut had lost once, it was time to let someone else try.

In the typical world where Tiamat and Bahamut are in an endless stalemate, I could see an evil dragon having the same kind of reasoning about Tiamat in addition to just raw ambition.

Yes, that's what i was thinking. Just wished there were an established deity here.

The specific deity you may be looking for would be Azharul, but as 6thLyranGuard said, canonically, Tiamat already defeated that rival.

However, if you want some more complex intrigue, you could have both cults worshipping different avatars of Tiamat, unaware that Tiamat is driving them to fight one another for her own reasons. This could be a metaphor for a "Two-Party System", where Tiamat knows that she can grow her power by making her worshippers wage war with one another to make them more capable, zealous, and devoted. It's easy to perform miracles, when you control both sides of a conflict.

Or maybe, when Tiamat absorbed her rival, the divine spark still exists within her and the other cult is worshipping that spark to keep it alive in an attempt to give it enough strength to assault Tiamat from within.

Both of those are cool ideas.

An overly powerful Draco-Lich maybe? 

Null could definetly work. Canonically I think he might be stronger than her but there's not problem just making them equally powerful