Overall of course I completely agree with PP. Do not rush your child and do not let his own enthusiasm rush him too much. I have seen talented boys rushed with skipping levels and getting lots of privates (more hours training) so they could skip levels, and in the end it can really backfire. They may not have the maturity to handle the longer training hours or competition pressure and the result is behavior problems at practice that become a frustration to the coaches and the other boys, and ultimately they end up either leaving the sport or eventually they stay at the same level for as long as needed for them to do what they have to do, so the push into the sport faster makes no difference or makes things worse.
But, your son's birthday means that once he starts competing, he will always be competing in an age division that is up a year from his actual age for most of the season. So that got me thinking. *I hope someone else will read this next part to make sure I am getting this age thing correct.
The age cut off is May 31 of the competition season. So if (in a fantasy world) your child was allowed to compete next season (2017-18) he would be considered 6. But according to folks here, as pp says, child must be actually 6 before he can compete. So in that case, here is how I would figure it.
Your son turns 5 in April 2017. So, he will be 5 and not be able to compete for most of the 2017-18 season...but he will actually turn 6 before that season is over, possibly- Right? So, could he compete at a meet that happened after he turns 6 (if there was one around that late?) Could he compete at his state meet if he has not competed all season but is 6 by that meet? I do not know, but it might be something to ask coach about. This is assuming this is something you and he want to do, he has the maturity to handle it etc. Again there is no reason he needs to start competing sooner rather than later.
In any case, so then, he turns 6 in April 2018. He WILL be able to compete the 2018-2019 season. And he will be considered a 7 year old for that season's competitions, even though he will be 6 for most of the season.
So then the question would be, given his interest in more hours and early start on preteam, maybe it would make sense for your son to skip Level 4 and compete as a 5 his first year. Also his coach could take into account the new bonus/no bonus divisions in each compulsory level. Maybe he could be a 5 Division 2, which means no bonuses? Or he could compete 4 Division 1 (with bonuses.) In other words, there are many options.
As far as time in the gym and cost, given your son's interest I see no harm in adding another day of a gym class at this age if that is available and see how that goes. Many boys preteam programs the kids go 2 days a week and that is 4 and 5 year olds.
For hours going forward, you will have to talk to your coach about how they do it, but in my experience (at one gym) kids training Level 4 go 2-3 days a week for about a 2 hours practice, and the big jump happens when training Level 5 where they start longer practices (2.5 to 3 hours) for 4-5 days a week. And it increases more incrementally from there. I do not know how costs are done for preteam at your gym, but you will want to ask how much team training tuition is at your gym. It is not usually just multiples of what rec costs, as that would quickly become impossibly expensive. However there is no getting around the fact that gym team is an expensive sport. On top of practice tuition, there are the meet costs/cost of competing and that is handled differently at every gym, and includes many things you may not think of like per diem for coaches and coaches salaries etc. aside the cost of meet registration. And then you have to think about uniform cost, cost to your family for meets for travel, food, hotels possibly, and meet spectator admission.