Here's an interesting possible attack on this problem. Let N be an integer with a ton of representations of the form a^2+b^2-ab (equivalent form: u^2+3*v^2). For example N=7*13*19*31*37 works well, (product of K primes that are each 1 mod 3 will have a number T of reps exponential in K). Now within the set of vertices of the equilateral triangle lattice of sidelength=1, consider the distance=sqrt(N) graph. This graph is countably infinite and regular with (arbitrarily large) valency T. Now ask your computer: what is the chromatic number of some finite subgraph of it? That will be a lower bound on the chromatic number of the plane distance=1 graph. The analogous attack can be done using any underlying 2-variable integer quadratic form (not necessarily a^2+b^2-ab), and which quadratic form and which N would yield the best results, I do not know. Because of the large valency it seems plausible you might get a large chromatic number. Unfortunately the specific N just suggested would yield a rather large graph... your poor computer would be laboring trying to find its chromatic number...