The WiFi antenna on the router/booster does not transmit in a perfectly spherical pattern. It is directional and irregular to some extent in all 3 dimensions. If the TV's WiFi antenna is close to the booster it could be in the weak part of the signal because signal pattern hasn't had a chance to spread out yet. There can also be some reflective interferance being that close to the source. Moving the booster further away as Sony recommends is worth trying.
Is smart steering turned on on the router? It's possible the TV is flop-flopping between 2.4 and 5 GHz bands which will cause the drop outs. This flip-flopping may be caused or exacerbated by the poximity issue. Try turning off smart steering if moving the booster doesn't solve the issue.