+1 Budget build + Brooks and VO bars?

+1 4mm difference in height, which is less than 1/4 inch (a little over 1/8th inch).

Does bike have steel rims? I am changing them out for sure. Myself, I just pick up used wheelsets. 27 inch wheels tend to be cheaper (not always the case). Advantages of 700c are minor, with the possibility of brake reach issues. 4mm is not much, but if pads are already bottomed out?

Often, used wheelsets come with good tires, tubes, skewers, and a freewheel or cassette. My last 27 inch wheelset pickup was earlier this year, came with good tires and tubes, skewers, PLUS four brand new in the box Michelin tires, and a good freewheel. $30. Wheels came off a nice Cannondale touring bike with light use, seller was upgrading to 700c. Four extra new tires are worth more than that, so I am ahead already. I'll be putting them on the Trek 520 I picked up Saturday (non original wheels on it). And then those wheels will go on a SS. The reuse cycle continues.

And often, wheels come in the form of a complete bike.