I think one way is to just subtract 6.9 from your y and fit your regression to new y's:
ystar = y - 6.9
Put this ystar in your J data matrix, call new data matrix with ystar, Jstar
Then fit the regression to ystar without a constant using the option IncludeConstantBasis -> False
LinearModelFit[Jstar, {x1, x2, x3, x4}, {x1, x2, x3, x4}, IncludeConstantBasis -> False]
Your "constrained" model is then the coefficients from above regression with the intercept 6.9 as required.
