Pretty far
First off, which LT trailhead at the base of Pico do you mean?
The most direct trail route down from Pico will be the 'old' LT aka Sherburn Pass Trail down to the Inn at Long Trail. That's another mile east on Rt. 4 from the current LT crossing. However, to get back to the current LT (and down) you will have to backtrack (but only 0.4 miles) from the Pico spur, unless you are planning a bushwhack there. Maybe the fastest way down from Pico toward where you want to start road walking would be via ski trails.
It is far enough from any of these trailheads back to Brewer's Corner (and the base of the Bucklin Trail), that I would look for an alternative. Even if you don't mind the road walk, the bit on Rt. 4 to get over to Wheelerville Road would not be fun, I think.
Backtrack: One alternative, after Pico, is to simply head back toward Killington, and go down the Bucklin. Might be a longish day, but quite straightforward, at least.
Catamount: There is also a XC ski trail between the current LT crossing at Rt. 4 and Brewer's Corner. It does not follow as flat a route as you imagine it could have. If you want to try this (a little risky if you ask me), then I can give you a little more information on it. I would definitely do this section first, not last.
Lost: If you really want to get fancy, the 1997 topo map shows two trails/ forest roads coming right up to the Ram's Head - Pico ridge traversed by the new LT route. I recall casually looking for these when I passed on the LT last fall, but I don't think they are obvious (unlike the path up from the Mendon forest road). Via a bit of a maze, these routes join the Bucklin Trail down low. I am not recommending this.
Tour de Mendon: If it were me, I'd simply leave my car across from the Inn at Long Trail, then ride my bicycle out Rt. 4 and in to Brewer's Corner. The Rt. 4 part is pretty much all downhill, and early in the morning biking on Wheelerville Road, I'd have a good chance of surprising some wildlife along the way. I always bike to the trailhead, walk to the car. Again, I don't leave the tricky part for last.
Winter: Actually, I am looking for the optimal WINTER route to combine Mendon, Killington, and Pico. I think it will be up Mendon via the forest road / short bushwhack, then continue up that road to the LT to Killington, then via Sherburn Pass Trail to Pico spur and out to the Inn at Long Trail. We will have two cars. Probably easier, but riskier in winter, would be this in reverse, ending at Brewer's Corner. Long day either way.