The short answer
Were Landa and Hellstrom lovers? Is there any proof?
There is no proof and I don't think this is the case.
Why would he kill her in such an intimate fashion otherwise?
Landa killed Bridget because he wanted to (rather than needing to), and he did it quietly so as not to harm his changes of defecting as they were his only way to avoid being on the losing side of the war.
The long answer
Link to the script for references
1. Landa does not defect out of the goodness of his heart
Landa is a really unusual character. He is charming and seemingly kindhearted, but there is a ruthlessness underneath that guise.
Your question is effectively contrasting the murder (bad) with the defecting (good), but that's not how Landa's mind works. He wasn't considering defecting because he had a change of heart or suddenly realized that the Nazis were (surprisingly) the bad guys. He was considering defecting because he thinks they're going to lose the war.
2. Landa is a textbook psychopath/sociopath
Above all else, Landa lacks empathy. He reveals as much when he discusses his nickname and job:
[I'm aware] that they call you, "The Jew Hunter".
Precisely! Now I understand your trepidation in repeating it. Before he was assassinated, Heydrich apparently hated the moniker the good people of Prague bestowed on him. Actually why he would hate the name, "The Hangman", is baffling to me It would appear he did everything in his power to earn it. But I, on the other hand, love my unofficial title, precisely because I've earned it.
The rest of this conversation is Landa justifying why hunting Jews to extermination is philosophically sound, using the Socratic method by getting Perrier to agree with everything step by step.
This entire scene reveals Landa's charm and intellect, but it also subtly reveals how much Landa lacks empathy or compassion.
Not too dissimilar from how Hitler has been argued to be a great orator and thus was able to sway public opinion, Landa is showing a similar trait whereby his charming character and generally kind demeanor hides the cruelty behind his actions.
It's arguable whether Landa's justification for hunting Jews are his true beliefs or simply claims to justify his job. He sure claims it to be true, and because he is such a great liar we're unable to definitively see which parts of his statements are lies and which are true.
But regardless of whether he lies about it or not, Landa is a narrative evil. If he is a genuine antisemite, this is obvious. If he is not a genuine antisemite, that means he partakes in genocide for reasons he doesn't even believe in, which requires a tremendous lack of empathy.
His eventual willingness to defect (even if for selfish reasons) suggests that Landa is not a genuine antisemite, or at the very least has no undying loyalty to that ideology.
3. Landa killed her because she is his enemy and he wants to punish her
When he kills her, he already has all the proof he needs that she is guilty of collaboration with the enemy (she admitted it in the end).
Just as how the Jews are perceived as enemies of the state and Landa hunts them to extinction, Bridget has declared herself an enemy of the state and this makes her no different from the Jews as far as Landa is concerned. Her murder is as justified (to him) as the murder of the Jews he hunts.
Let's look at this from the moment just before he murders her but has proof.
He is already aware (due to Brad Pitt's pitiful Italian skills) that the Americans have infiltrated the gathering. He's aware how vulnerable they are as all high ranking officials are present here. Assuming his intelligence displayed in the movie, we can infer that he has puzzled all the pieces together to know that the attack is imminent and cannot be avoided anymore.
Landa had already been contemplating defection
Not quite. This is where he decides to defect, as he realizes the walls are closing in around him.
Notice his actions right after the murder:
Inform The Fuhrer the audience has taken their seats, and we're ready to begin. Step one, in Hans' master plan, done. [He then dials another number]
There is a lot to unpack here:
- He allows the precedings to continue, even though he has enough justifiable evidence that the evening's security is compromised and an attack is imminent.
- He has a master plan - which is later revealed to be defection
- It is never confirmed but later implied that this second phone call is specifically to initiate his defection
So if he chooses to defect, he should embrace Bridget and work with her, you'd think, right? But he doesn't, because he never had a change of heart. Defecting was purely a matter of self preservation. And regardless of his imminent defection, he was still faced with Bridget who has openly declared herself his enemy and has already helped the allies to kill nazis.
This violence is unnecessary for performing his duty since the cinema is packed with the German military and security.
Quite the opposite. Killing her quietly was the only way to kill her. If he had done so publicly (or had her arrested) for collaboration, it would have hurt his credibility as a defector.
In short, he killed her because he wanted to (more so than actually needing to), and he did it in a way that did not harm his chances of defecting and escaping the losing side of the war.