I'm talking about a team actually working with a player to try to achieve a more favorable result for the team.
I strongly doubt it. If the kid is 'that' good, somebody will probably grab him before Team 1. A couple of interviews won't out balance highlight reels, prior interviews, etc. Maybe drop a spot or three (depending on the quality of the draft too), but if Team 1 had say the 8th pick & he was projected prior to the draft to be #3-4...probably not.
Then too you are expecting a LOT of people to keep their mouths absolutely shut for years. Probably at least a dozen for Team 1 (execs, scouting dept, coaches, interviewers, lawyers (most likely), secretaries doing all the paperwork, etc. Then the player, his agent, probably his parents (wondering why he wants to go later in the draft...), girlfriend, etc.
One person tipsy in a bar trying to impress someone with how important they are. Girlfriend getting dumped for someone hotter. Secretary pissed they didn't get an expected raise. Etc. You're relying on people with grudges or being stupid not to do or say anything...forever.
The cost:benefit ratio just isn't there for that level of tampering. Especially when it's unknown if he'll even work out & the others around him in the projected draft order probably aren't 'that' different than him all things considered.
Much safer to just try to trade up in the order to get him if you want him that badly.