どうしても入り方にこだわりたいのか...
I'd like to know how 方に works here. I don't think it's read 入{はい}り方{がた}, 'time of setting', given the context I have. More likely it's 入り + 方に, since in the story the speaker seems be coerced into entering some kind of contest.
I found an example on Tatoeba.org of 方{ほう}に translated as 'made for':
彼{かれ}は戸口{とぐち}の方{ほう}に進{すす}んだ
'He made for the door'
Maybe that use of 方に could fit into the sentence in question...
Pretty sure it's 入{い}り方{かた}, which here means "way/means of entering [the contest]". And こだわる here is probably "insist on, make fuss about". So, I get something like:
"Do you really want to get me into that contest no matter what?"
Check a grammar book about the ～方 suffix.
First of all, 「入り方」 is read 「はいりかた」 --「かた」, not 「がた」. It means "how to 入る".
Although OP has not provided enough context, I am going to jump the gun and post an answer.
「入る」 here looks like it is being used for its slang meaning of "to start a performance". It is used rather heavily in show biz since many performers think the beginning part is important for the success of the rest of their performance.
「～～にこだわる」 means "to stick to one's way of doing ~~".
Thus, the sentence in question would mean:
"Do you absolutely have to (or "want to") stick to your own way of starting your performance?"
