This is a very marginal spot, right on the borderline - and is thus
heavily read dependent. As such, with no reads there is probably not a great deal in it between piling the money in and making a tight fold (obv the only thing not to do is to call here).
In this particular situation I would tend towards piling the money in though. This is based on having played with Neil quite a bit at padooki over the last few months and noticing that he is not as cognisant of positional considerations at padooki as he should be. Added to this is the fact that he is drunk so I would guess that he is a little more likely than usual to be wide for value here.
These sort of spots (i.e. having a made padooki with the ability to break and draw smooth) are unique to padooki-type games, and analogous situations do not occur in games like Hold'em or Omaha. Essentially your 4bet is a
hybrid bet, a sort of merge that does not fit into a precise category of either (semi-)bluffing or value-betting. Sometimes you fold out a better made padooki like a rough J or T (you have bluffed out the best hand), other times you get action from a smooth draw that wants to gamble with you (you have 4bet for value), and other times you get called by a better padooki but still have a decent chance to suck out (and now you must rely on the 'semi' part of your semi-bluff).
Three important points to make though:
1. If both Pete and Neil get all-in with you here then you
must break the Q, even if Neil draws one in front of you. You are a dog vs two one-card draws and have around 33% equity. Of course, if you
knew that Pete was also drawing then you could pat and take your chances since you do have about 33%. But the chance that Pete might be pat behind you makes this a
clear break.
2. If you end up HU with Neil and he pats in front of you then you should usually break the Q immediately rather than trying to beat him in a game of chicken. I won't go into the maths on this, all I will say is that it is a serious mathematical mistake to give up one draw's worth of equity on the off-chance that you can 'trick' him into breaking a J or a T.
3. If HU with Neil and he draws then obviously you pat. However, if he pats on a later draw then you should break - barring a ninja read that he is putting a move on you with a flush/bigger padooki. It is a very bad play for him to pat as a bluff in this spot for several reasons, the main one being the strength of your range. So if he randomly manages to get you to break by patting a small flush (and you fail to outdraw him when you break) then you can just console yourself that he made a mistake against your range, and simply got lucky to run into the bottom of that range.
N.B. If you had a hand like

it is a MUCH easier 4bet. It might not look like too much of a difference, but replacing the 6 with a 4 makes a significant difference to equities in those situations where you end up breaking. In general, the majority of players do not appreciate the importance of drawing smooth at this game. In fact, if your hand was

then I would recommend almost always folding it. Like I said, the

is right on the borderline.
Think that is all I am going to say on this hand. Don't want to give the shop away for free at padooki.