What you say is true. You may submit your application for your visa whenever you want. You probably do not have to show bank statements to the Embassy, just an apostilled letter documenting your federal/private pension. SD will ask you for proof of pension payments.
The problem you might have is that before you can submit your visa application to the Embassy, you have to gather all the documents so you can submit them. When you get your criminal record check, there will be a date on it. Let's say your get this in December 2015 (one month before you leave for the DR). This document is only valid for submission in SD until May 2016. If your long form birth certificate is ordered in November, then it is only valid until April.
You submit your visa application to the Embassy before you leave the US in Jan. 2016. They process your application and ask you to submit your passport for your visa stamp say early Feb 2016. For the sake of argument, let's say the visa is issued on Feb 10 ( you have to get your passport to the Embassy). The residency visa is valid until May 10.
You have to wait until the end of March for your third bank statement. Sometime before the end of March, you get your passport and documents back (how you get them here is up to you). Between the time you get your passport back with the visa in it and the end of March when you have your third bank statement, you need to leave the DR to clear your entry as a tourist and pay the overstay fine. Then you need to renter the DR using your residency visa. Where you go is up to you but you do have to leave and go somewhere.
It is now early April 2016. You have your visa, your passport, your documents, your three bank statements and you have re-entered the DR for the purposes of filing for residency using your residency visa. You run like 7734 to your lawyer and give them everything you have. They run like 7734 to SD and start the process of getting these documents submitted to Migracion. Sometime in the next 30 days your criminal record check will expire and will no longer be accepted. If your birth certificate is dated before December as noted above it will expire sometime in April. If there are any delays, your lawyer gets sick, Migracion doesn't like something or wants you to get something else or change something, you're running out of time. Your visa may still be valid but a document might expire.
Your time constraint will probably not be the visa itself expiring, but the documents you will have had to order, receive and apostilled before you leave the US in January. If you miss the timing, or if there is a delay in issuing the visa by the embassy and it's always possible that your application acceptance in SD could be delayed, you might run out of time. If this occurs and a document expires, by the time you get a replacement, your visa will have expired and you still won't be able to proceed.
When I did this, I ordered the documents 2 months before I planned to enter the DR on the visa (entry in Sept.). I got them translated, apostilled, notarized etc so that I could apply for the visa in early August. Got my visa towards the end of August (about two weeks before I was scheduled to come to the DR.
This way, my lawyer had almost the full 90 days to do her thing. None of my documents would expire until after the visa did. Good thing too, because there was a bunch of running around that had to happen and of course there were some rule changes that necessitated some more paperwork involving the DR Foreign Affairs Office.
I'm just saying that you need to be very careful that you don't get caught up in a situation that precludes you from submitting your application just because you want/need to be in the DR in January instead of say April when you could arrive with all the documents you need with lots of time before they expire, a full 90 days on your visa and of course 90 days to counter any asinine moves Migracion may or may not choose to undertake. It would be pretty crappy if the rules changed again while you were here in the DR with time ticking away on your other documents while you wait for a third pension payment. As you probably know from what others have said here on DR1 - when it comes to Migracion, nothing ever seems to go exactly as planned.