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English uses this negative “Isn’t it?” quite often, also (じゃないか). It’s entirely rhetorical, and despite seeming like it’s asking for confirmation, it kind of implies the person using it is fairly confident in what they are saying.
Literal translation:
“Ehhhhh!? How/why!? More than yesterday, becoming more, isn’t it… How/why!?”
A bit more naturally phrased in English:
“Ehhhh!? How/why!? Hasn’t (the amount of times I was teased) become more than yesterday?.. How/why!?”
もう = already
イヤ = disagreeable, unpleasant
だ = copula
もうイヤだ is literally “This is already unpleasant…”, but often gets translated as “I don’t want to do this anymore” (since the same vibes are there, even if the exact words aren’t)
筋トレなんて = this is the “something such as” なんて, and it’s being used because he is displeased about it, so he uses the なんて to give that diminishing vibe. I’m afraid I don’t know the word dispregiative, but I assume this meaning of なんて is what you are referring to when you say that.
I think the line break matters, so the もうイヤだ stands on its own a bit, and the 筋トレなんて is just a follow up to clarify the subject. “This is already unpleasant/I don’t want to do this anymore. Such a thing as muscle training…”
If it were in “proper” order: 筋トレなんてもうイヤだ…
“Such a thing as muscle training is already unpleasant.” or “I don’t want to do such a thing as muscle training anymore.”
The reason I think the line break matters is that いや is a な adjective, so if it were to modify the 筋トレなんて, it shouldn’t be using the だ copula to do so. Maybe somebody disagrees with me on this. Regardless, the meaning of the sentence doesn’t change.
明日こそは = Tomorrow, with emphasizing こそ + は topic marker
からかわれない = to not be teased
ように = in order to; so that
気をつける = to be careful (る is dropped for purposes of connecting to ねば)
ねば = short for ねばならない. ねば is actually something archaic, and is really only used for purposes of making a joke a bit. It’s overdramatic because it’s old fashioned. なければ would be the modern equivalent. This ね is the realis form of Classical Japanese auxiliary verb ず, and the ば is the same conditional used in modern Japanese. But either way, it has the same meaning of なければ: “must do～”
Translation:
“Tomorrow, in order to (ように) not be teased (からかわれない), I must be careful (気をつけねば).”
I would probably rearrange the phrasing in English to sound more natural:
“I have to be careful so that I don’t get teased tomorrow.”
But leaving it as the slightly wordier/more formal sounding phrasing might be a better way to get the joke across about the overdramatic usage of ねば. 