The barrel will handle any pressure that the factory loads develop, that would include a rebore from 30-30 to 375 or 375-06 for that matter.
The suggestion was only to get true 375 Win bore dia. The 38-55 and 375 are not the same dia and that was why Winchester made the bulets a true .375 for the 375 caliber. The 38-55's have always been at least .376, usually more like .378 or larger, in most guns thus chambered.
Your target model will handle the pressures of the 375 no problem, but the bullets may not. There is a decided lack of jacketed bullets in the 38-55 dia, but not so in the .375. Once you push cast bullets over 2,000fps leading can become an issue. Some have this, some do not, but 2,000 is the general rule.
I am using load date from Hodgdons books and with a 200gr jacketed bullet, it pushes the high 2400's and low 2500's from my 20" barreled levergun. Start pushing cast bullets that fast and you leave a lot of the bullet in the barrel.
By the way, there'd be more steel left around the handi 30-30 barrel, bored to .375, than there is on my factory 375 Win barrel of my 94 BB.