I know "I had had to eat burger" is the "past perfect tense" and "I had to eat burger" is "simple past tense" and we normally not use "got" in the same sentence structure as in "I had got to eat burger" but if i use "got", Would the sentence be considered "past perfect tense" or " simple past tense"?
I think that you are asking two different questions. The first is whether you can use the "to have" + "got" construction in different tenses:
Past: I had got to . . .
Present: I have got to . . .
Future: I will have got to . . .
The answer is no, the past and future versions are not normally used. Only the present version ("I have got to . . .") is used, and even then it is considered quite informal (as you point out).
The second question I think you are asking is whether this would be considered the simple present tense or present perfect tense. You might get some different opinions about this, but because it follows the typical structure for the perfect aspect ("to have" + past participle), I would describe this as present perfect. (Note that "gotten" is the more usual past participle of "to get", but "got" is used exclusively in this construction.)
Finally, your sentence is missing a determiner for burger. It should be:
"I have got to eat a burger." (Or "that burger", "my burger", etc.)