All four are acceptable (with one change, below) and fairly interchangeable. They're all casual ways of saying that you intend to eat.
There's one issue: "I get going for dinner" is not idiomatic. It's confusing, because we're already using both "going" and "get" in "I'm going to get dinner." But "get going" is its own phrase, meaning "start going," and using it with the simple present tense is awkward. "I'll get going for dinner" would be fine.
All the options have essentially the same meaning, but there are slight, subtle differences of tone:
- I need to have dinner now. This is very straightforward and descriptive. It talks about your "need." The tone is a bit blunt and dismissive; it might be appropriate when someone has taken too much of your time and you are not trying to be too courteous.
- I'll get going for dinner. This has a pleasant, informal tone. The "get going" would make it especially appropriate if you need to leave the room, perhaps to drive somewhere.
- I gotta go grab a bite. This is the most informal. Yes, "grab a bite" can imply that the meal will be short. You could use this if you intend to be back in contact after the meal, and are suggesting that it won't take long (though it wouldn't be inappropriate even if you're not coming back).
- I'm going to get dinner. This is neutral in tone. "Get dinner" could also be used even if you're preparing it yourself, though if so, you could also use words referencing that, like "fix dinner" or "make dinner" (which imply that you will also eat it).