小島：塾なんかやめればいーじゃん
オレ家庭教師にしたぜ伊藤：うざいんだ 家庭教師って
部屋で2人っきりてのがどーもな
I am struggling with the bolded phrase, I think he says something to the meaning of "What are you going to do with just two people in a room" but think I may be missing something because I am not sure if どーも can be interpreted that way since I can't find a use case.
て I think might be short for という but in that case I would expect って so I am not 100% sure about that either.
The dictionary form of:
「～～てのがどーも」
would be:
「～～というのがどうも」
In this context, 「どうも」 is used for an ambiguous expression of a (somewhat) negative feeling. It is similar in meaning to 「なんだか」、「ちょっと」 or 「なんとなく」. This 「どうも」 is used quite often in informal speech as it saves us the trouble of selecting adjectives, verb phrases, etc. with more precise or concrete meanings.
Here are the definitions for this usage of 「どうも」 from two monolingual dictionaries. The key phrase for both is 「はっきりはしない」.
広辞苑: ②はっきりはしないが、どことなく。何だか。何かしら。
明鏡国語辞典: ❶ はっきり断定{だんてい}できるわけではないが、なんとなくそのような状況{じょうきょう}だと感{かん}じられる気持{きも}ちを表{あらわ}す。なんとしても。どう考{かんが}えても。なんだか。
For a translation, I might use "dunno how to put it, but it's kinda meh (to me)" or simply, "It's kinda..., y'know."
Thus, this speaker is implying that the idea of being alone with a private tutor in a room does not appeal to him.
Finally, 「て」 surely means the same as 「って」 and 「という」 here. In colloquial speech, some young people use 「て」 instead of 「って」 in recent years.
