I think with this stochastic process it might not be so simple to get the individual log likelihood contributions and maybe the brute force method you show above is about all one can do. (I'd very much like to be wrong about that.) Here's why I think that way:

Consider the difference in the log likelihoods for a sample size of 3 and 4 with values x1, x2, x3, and x4:

Expand[PowerExpand[LogLikelihood[FractionalGaussianNoiseProcess[0., 0.15, 0.9], {x1, x2, x3, x4}]]] - 
 Expand[PowerExpand[LogLikelihood[FractionalGaussianNoiseProcess[0., 0.15, 0.9], {x1, x2, x3}]]]

1.4035257547656506 - 1.0886020257023077 x1^2 - 1.3775254495136622 x1 x2 - 0.4357828479222121 x2^2 - 8.76194837289972 x1 x3 - 5.543718726413118 x2 x3 - 17.63080939515791 x3^2 + 15.05161120348482 x1 x4 + 9.523212799282188 x2 x4 + 60.57376211881078 x3 x4 - 52.02796671141442 x4^2

We see that x4 isn't isolated by itself and all of the previous values are included in the difference in the log likelihoods.

If 0.15 and 0.9 are replaced by unknown values (and PowerExpand is told that all the values are Reals), the resulting equations for just a sample size of 4 are way to lengthy to show here. Maybe someone else more familiar with this stochastic process can shed more light.