1
The Chicago Manual of Style recommends:
[In dialogue,] if the pronunciation of the abbreviation would be understood by most readers, you can write it as you normally would:
“I’m going to the store, a.k.a. the shop,” my roommate informed me. “Would you like me to get you anything?”
The same goes for an abbreviation like “Ms.” or “NASA” that would be understood (and said) as a word rather than as a series of letters.
Since in American English it is common to address people by their initials, and native American English speakers will know how to pronounce the initials TP, you can use that version of your character's name in both the narrative (outside of dialogue) and in dialogue:
TP said: "Hello, John!"
John replied: "Hi, TP."
2
In American English, I would normally use the spelling recommended by the Merriam-Webster dictionary. Merriam-Webster gives tepee as the main entry, teepee as a "variant spelling of tepee", and tipi as "less common spelling of tepee". I would therefore choose tepee.
You may have reasons to use another spelling (e.g. tipi is the Daktoa and Lakota spelling). I would use the spelling common to your readers if you want to use tepee as a common word of the American English language (as in your case, when one character makes fun of another's name), and tipi in an anthropological or ethnological context (e.g. in a story about Native Americans).
3
Whenever one character means to say tepee, use that word:
TP said: "Hey, John!"
"Hi, tepee," John replied and snickered.
TP rolled his eyes.
Your narrative ("snickered", "rolled his eyes") will make it clear to your readers that your characters understand what word is being said.