The density of white oak is about double that of WRC, so each piece made from white oak will weigh twice as much as if made from WRC. Probably a noticeable difference.

True BUT white oak is also a lot stronger than WRC. You can use thinner and/or narrower stock to save weight and get the same strength.

From the USFS Wood Handbook "fpl_gtr190.pdf" (truly a bible for designing and building with wood):
The Modulus of Rupture (static bending) is 100% greater for whit oak compared to WRC:
7,500 lbf in–2 (RWC) vs 15,200 lbf in2 (wt oak)

The tensile and compressive strengths are all much higher for white oak.

Getting into the weeds...

The bending strength is determined by the cross section of the member, the moment of inertia, which is a function to the power of four. Changes to width, while keeping the thickness the same, yield linear changes in strength. Reducing the width by 1/2 reduced the strength and weight by 1/2. A piece of 1/2"x1/4" WRC would have roughly the same bending strength as a piece of wt-oak that is 1/2" x 1/4".

Changes to the height (thickness) for the same width are tripled though. Half the thickness yields a moment that is 1/8 the original. So to for half the moment you'd want about 80% of the thickness. Thinning the white oak stock down by 25% (say from 1/4" to 3/16") saves a 1/4 of the weight increase with nearly equal strength of 1/4" WRC stock.

Be careful changing both the width and thickness!

You have to be a bit careful with these reductions though. The strength numbers are based on clear, straight grained, "ideal" lumber. Any runout or grain changes will impact the bending strengths. Lumber weights can also vary widely even within boards cut from the same tree.

Hope it helps. Have fun building!