First unlike others here I don't think getting your owner draw in the following month is odd at all. That is the way my property management companies did and do it. My background is commercial real estate and that is a normal way of doing it there. It took explaining it to some residential clients at first but there are lots of benefits to it.
As far as the book keeping on your end, just keep it simple. Since you are splitting up your property books, (your property manager doing part and you doing part) it will be a little messy and require adjustments at the end of the year. When you get your owner draw input it in the books at that time and don't worry about back dating it. Make sure to ask your accountant how they want you to input it. It shouldn't be input as income as it is not income, your property managers already input it as income so that would be counting it twice. It is an owner draw so that means it goes on the balance sheet and not the income sheet. Your accountant will need to merge the two reports together at the end of the year for taxes, thus you should check how they want it input, so that it takes them less time and thus costs you less money. There is a chance they will say just to send them the P&L from your property manager and the expenses that you also paid and completely ignore the rest of your books so it doesn't matter.
Recording it in September or October will not make any difference on your trying to get a feel for your monthly cash flow, as long as you are consistent in the way you do it. If your owner draw coming in every month is $1000 and your mortgage going out is $700 and that goes on for 6 months, it doesn't matter if you show it in the prior month or this month; it will still show $1000 in and $700 out, thus you will be able to see and understand your cashflow. Hopefully this makes sense. BTW one of the benefits of the paid in arrears is the consistency. You should be getting your owner draw at the same time every month regardless of if your tenant paid 5 or 10 days late or not.