P. S. I guess “ウチで” might mean “I”, many people say “ウチで” as “I”, especially among young girls. But in this case, I don’t think it’s I.
I am going to elaborate A.Koba's P.S. comment about the usage of "ウチ" as the 一人称単数形の人称代名詞「私」 first person singular form of a personal pronoun "I,my,me" for those who are not familiar with the word.
In the standard Japanese, うち means house (maybe except for the usage of うちの in うちの会社
my company and うちの娘
my daughter).
うち however has always been used as the first person singular form in Kansai dialect, and it is interchangeable with 「私」.
Many think that TV has spread the Kansai dialect nationwide, and it is now a trend for younger females in Kanto/Tokyo area to use use うち the same way as A. Koba mentioned.
By the way, I was not commenting on the usage of ウチ as the first person singular in my previous post but rather was on the suffix .To translate it as ”I” instead of "My company", I just thought うちが sounded a bit more natural than うちで in this particular sentence, but it may be fine just as it is.
Also, 「よって」(which means から) in 面倒みるよって seems to be the Osaka dialect (part of the Kansai dialect), but personally I am more familiar with such phrases as 面倒みまっさかい、面倒みるさかい(Osaka) or 面倒みるし(Kyoto).
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In my previous post, I mentioned the phrase did not sound polite enough to talk to a customer.
Instead of "載せかえまでチャンとウチで面倒みるよって", "載せかえまでチャンとウチで面倒み
させてもらいますよって" would sound more normal when you talk to a customer.