So, you know the way to Mud sump. Good.
Mud sump can be done with a few inches airspace, slowly!, helmet off, on your back, head first, nose scraping the rock roof - don't make any waves, however!.... It's no longer than the airspace you can see but because it is a bit of a U-tube and also awkwardly shaped, it cannot be free dived in the "usual" fashion (i.e. as with sump 1). Bailing it doesn't take too long if there's more than two of you (two people can bail it in 30+ minutes) - but don't leave any bailed water in the lower dam since it will damage it by slowly leaking back out. [There are lots of signs explaining how the system works].
The troubles can usually be bailed by setting up the siphon system in situ (if you can work out how to get it going...) and having a quick nose around down in Swildon's 4 (a bit tight down Blue Pencil but I can do it and my chest measurement is "Johnny Pies, XXL FB") while the level drops. The troubles can be done in a similar fashion to Mud sump (helmet off, on your back etc.). You will meet further ducks which can be more quickly bailed if they require it before getting to my nemesis.... birthday squeeze (approached by taking the obvious left turn, up a v. short climb/step, into a downhill passage where the choked stal obstacle is to be found). I always strip off my belt, oversuit and helmet to do the squeeze and although I'm only in my furry + pads and boots with a Petzl tikka to see the way, even with my chest flattened and forcing myself through by sheer leg power it's easy to crack your ribs doing this. I hate it. I have managed it four times and the last time did my ribs (again) so have not returned since. If you are a big lad (i.e. 50"+ chest) you may not get through and this would mean a return out the way you came.
An option is to go through sump 1 down to the landing and up towards Vicarage passage and then start doing the round trip in reverse; get through the nice flat "duck" and up to the birthday squeeze; strip your gear down and have a go and pushing yourself up through the obstacle - if you can do it from this side you can definitely do it from the other; this quick test can be done on a recce trip which shouldn't take you more than an hour (or so) all in - it beats having to do the round trip from the "correct" direction only to have to beat a retreat upon failing to negotiate the B'day squeeze if you're a large person.... a journey which is considerably longer than doing the round trip!
Possibly the best plan is to expect to get to the B'day squeeze but have to return back the way you came; this way, if you DO have to retreat you will only be doing what you expected anyway and it will not seem like a defeat (if there's two of you, allow at least a couple of hours for the journey - depending on your speed, perhaps 3 might be more realistic). The Round Trip can be done at a leisurely pace and encompass a visit to Swildon's 4 streamway in 4-5hours if you're in a group of 2; there are some famous claims that the R Trip has been completed in under an hour (probably true) by cave sprinters - they do exist!... I've seen some - but you're going to be enjoying the journey so it's best to allow LOADS of time to do it. If you're in a large group (i.e. 7-8 students) then perhaps you'd better set aside a whole day! (I bumped into a group last year who were trying to find Tratman's Temple and had already been underground 4 hours! - after giving them directions I saw them the next day and I recall they said it'd taken them a further four hours to complete the trip).
For the record, I wear an XL warmbac suit so if you're larger than that you probably will be unable to get through. Also, the problem isn't your gut/belly (perhaps you've got a well defined six pack, perhaps not; mine's more of a party barrel) - it's your shoulders/ribcage; the obstacle is a smooth, small, inclined oval so as you breathe out you slide downhill and towards the right... this makes breathing in again v. difficult and can add to the sense of claustrophobia which you'll be experiencing. Just thinking about it and writing this makes me re-live it. Good luck.
P.S. Don't go up the approach to Candy's Calamity instead of the Greasy Chimney and don't confuse sumps 3 & 2 for sump 1! The way on at Blue Pencil is over the top of the smooth-sided trench and up towards the troubles - the "path" for the round trip is so well worn it's pretty obvious, really. Also, after getting through B'day squeeze (well done, that man!) and doing the final flat duck (puddle) you'll be heading downhill in a nice round passage, approaching Vicarage passage etc.. - don't miss the way on down to the streamway at the bottom of the slope (on the left through boulders) and then drop down to the top of the landing - don't fall off the ledge, use the handline rope to guide your descent.
You're an experienced caver.. you should be able to work out the trip from all the clues/smooth routes anyway. This advice of mine is probably definitive and you might want to ask around a bit from others who have done the trip - let them "eye you up" and see whether they reckon you'll make it through....
Just one final thing... sump 1 has become a bit "narrow" recently with pebbles getting washed in; it might be a good plan to drag some out with your leg before going through.