I definitely would not use another type of wood flooring. One of the first complaints you hear from potential home buyers on tv mention is the fact that the wood floors in different rooms don't match. I think it would look like you just ran out of wood. Somewhat of a hodgepodge. I'd use something completely different, like ceramic or porcelain tile. There are so many pretty ones out there.

Agree, agree. Take a look at some of the new porcelain tiles available.

Not sure if it matters, but this isnt a open floor panel. Its your typical old "siloed" house where the rooms all have doors. Would the fact the kitchen has a door opening serve to alleivate the "hodgepodge look?

We have some tile we like but cannot find it in the size we like. Here is my post regarding this issue. Please let me know if you have any ideas on this front?
https://www.houzz.com/discussions/herringbone-pattern-and-size-of-tile-in-small-space-dsvw-vd~384649

May I ask why don't you want to match the white oak in the kitchen? I'm asking because white oak is such a great, durable choice of hardwood for a kitchen that it seems odd not to continue it.

Figured cost would be higher relative to enginnered? In addition to having to seal it stan it etc.

My home has Hickory 5' enginnered wood and in looking to complete the master, I was blown away at how expensive it is.

White oak is abundant , good wood and easy to locate. Price will depend on grade of wood and if you are purchasing plain sawn quarter sawn or rift. Since your area is small price should not be substantial. Buying new unfinished and letting your installer knit in new with old and sand it will give you the best look in a small house.

If you need to save I see nothing wrong with putting plywood under the cabinets and only using the oak where it shows. I would raise the lower cabinets with the plywood under because I like keeping my countetops at least 36 inchs but some would leave it.

If you are in a vintage home, I don't believe that using a different wood in the kitchen is totally out of the question. In fact, it was quite common in older houses to have oak in the main public areas with maple in the kitchen. In my 1931 house, the kitchen floor is fir (original floor surface was linoleum, we finished the subfloor) and the rest of the house is quartersawn oak. It is an old house; kitchen is a separate room so why insist on having the same floor as elsewhere?

The best option is to get new wood and refinish everything to match. Just be sure to match grade, cut and species of wood which may mean paying more to avoid lots of shorts and uneven coloring.