I have some questions for you.
1. Have you changed your mind about buying a microphone? You can not use room correction software without a microphone.
2. What source are you using for your system? If you are using a PC, you can host the room correction on the PC instead of the sub.
When integrating subs with main speakers, you need (1) amplitude/volume matching and (2) time and phase alignment. The former is easier to do, it can be done by playing pink noise and using an SPL meter app on your phone, or even subjectively (turn the volume of the sub up and down until it sounds right). Doing it this way is very imprecise though, for that you need a sweep + mic.
The time and phase alignment is the tricky bit. For this reason, I think subs with built-in room correction is a terrible idea. Subs are usually delayed with respect to speakers, and adding DSP in the sub delays it even further. Typical sub delays with DSP is usually a minimum of 20ms, and up to 80ms. Adding even a 20ms delay to a sub is enough to make it sound slow and flabby. To achieve proper time alignment, it is the main speaker that needs to be delayed to match the sub. There are no subwoofer DSP modules I am aware of that are capable of delaying the main speakers.
Other ASR members have correctly pointed out that the sub does not give you slow and boomy bass, it is the very fact that bass itself exists. Bass has to be managed, there is no way around it. Of course, subs have to be correctly designed in the first place.
And lastly, there are two major factors working against you - your budget, and your knowledge. It is not my intention to disparage you, because all of us are in the same boat - we are all constrained by our budgets and our knowledge. But for subs, you definitely need a bigger budget and more knowledge. Subwoofer integration is not easy, and requires knowledge in room acoustics and whatever DSP software you are using. Fortunately, both can be acquired. May I respectfully suggest that you shelve this idea for now, save money for a bigger budget, and do some reading? Either that, or you need to revise your expectations. It is not unreasonable to have lower expectations, many people add subwoofers to their systems, fail to integrate it properly, but enjoy the sound anyway.