I mean to just apply again, but to use their priority service which is suppose to make things faster.
Paying for the priority service simply means that your application will be put to the front of the processing line, ahead of all the non-priority applications, but behind the priority applications that have already arrived.
While a previous visa refusal will most likely mean it takes a bit longer to process when they get to your application, using the priority service will mean you skip the waiting line - and the majority of the 3-4 month processing time over the summer was applications just sitting in line waiting to be picked up and looked at.
At the moment (November), if you use the priority service with a straightforward application (no refusals), then it has been taking about 2-4 days for processing and people are receiving their visas within a week or so of applying.
If you have a refusal, it may add 1-2 weeks onto that processing time, so if you do qualify and just didn't send the correct documents, then if you reapply now, you could probably have your visa in about 3 weeks.
If you don't use priority, I'm not sure what the current processing times are as everyone has been using priority lately, but I would guess maybe 4-6 weeks as an estimate?
The reason they say not to use the priority service if you have been refused before is because they cannot guarantee fast processing due to having to look into the refusal... so they don't want people using the service and then complaining that it took too long. However, I would say you might as well pay for priority since it should still be quicker than not using it.