How to solve an equation by substituting possible solutions?

Question

I wish to solve this equation for kappa, given a certain set of parameters (m and beta):

chi[x_, m_, beta_, kappa_] := (x*(1 - x))^beta*Exp[-(m^2/(2*kappa^2*x*(1 - x)))]
norm[m_, beta_, kappa_] := 1/NIntegrate[chi[x, m, beta, kappa]^2/(x*(1 - x)), {x, 0, 1}]
deltaM2[m_, beta_, kappa_] := norm[m, beta, kappa]m^2NIntegrate[chi[x, m, beta, kappa]^2/(x^2*(1 - 
x)^2), {x, 0, 1}]
M2[m_, beta_, kappa_] := deltaM2[m, beta, kappa] + 2*kappa^2
Solve[3.097^2 == M2[1.27, 2.56, kappa], kappa]

However, I am thrown this error message when I run the code:

NIntegrate::inumr: The integrand (E^(-(1.6129/(kappa^2 (1-x) x))) ((1-x) x)^5.12)/((1-x) x) has 
evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1}}.

It is clear that NIntegratecannot work since kappa remains a symbol. My question is if there is a way of solving this equation that substitutes in values for kappa to find a solution, so that NIntegrate can still work. These integrals cannot be solved analytically by Mathematica, from what I've tried.

I have found that using FindRoot[3.097^2 == M2[1.27, 2.56, kappa], {kappa, 0.1}] instead of Solve produces an answer, but I'm not sure why. I would also appreciate any different answers! Thank you.