Almost NEVER get an Sn2 with 3 Prime Carbon. Too much steric hinderance.
It's a primary halide with branching, not a tertiary halide.
To the OP: you cannot make a primary carbocation, as you did in your mechanism. If it were to go by ionization there would have to be a methyl group shift concerted with loss of the leaving group, similar to what happens in certain Friedel Craft's reactions.
This is an unusual compound, given the fact that it is both primary and yet very sterically hindered at the same time. I do not know if it would go by Sn2 or a weird Sn1 where the carbocation formation was concerted with an alkyl shift to give a tertiary cation as your first intermediate. Either way not a very good reaction, because it would take a long time and require high temps, etc probably.
Edit: the polarity and protic nature of the solvent will be critical in determining the mechanism. Polar protic will favor Sn1/E1, while polar aprotic will favor Sn2. Overall the higher the dielectric constant the more Sn1 will be favored, I assume also.