Easy. Glad you didn't take offense at my comment. Because of that, I will continue to help ya!
So, regarding your first question:
Yes, some things would change. Your diode has a lower forward voltage (my guess is around 3V?). Further, even though the diode current is the same, you won't be able to use exactly the same driver because of the voltage requirements (I believe LPC-826's can sustain around 500mA?).
Regarding fitting the LM317 driver into an Aixiz module... may I recommend just saying, "Screw it" to the Aixiz back end, stick the diode in the front end, and then just build the circuit in the large cavity that is the inside of the phaser? You have a lot more room for heatsinking and driver space that way, and that's how most of us do it (most of us have tons of back ends from those Aixiz modules

).
Buying a FlexDrive, while always an option, isn't worth it, in my opinion. They are expensive, whereas you can A) buy a cheapo LED driver and modify it to fit your needs from websites like DealExtreme and DinoDirect, or B) make your own LM1117 (a lower-dropout, smaller version of the LM317) driver and use that (which is the option I recommend).
Finally, alkaline batteries essentially suck. They can't provide much current (ESPECIALLY 9Vs), aren't rechargeable, and don't last long. They are about the worst type of battery in this hobby.
Therefore, I would suggest investing in some good quality rechargeable Li-ion batteries. They have a much higher nominal voltage (3.7V as opposed to 1.5V of a normal alkaline cell), can provide a TON of current, and are rechargeable (of course!). They come in various sizes, but some of the most common are the 10440 (AAA size), 14500 (AA size), 16340 (CR123A, camera battery size), and the powerful 18650, the batteries that power your laptop.
I understand that you want to save money, but in general, that's not going to happen with this hobby! It's expensive, so if you aren't willing to spend a bit (or figure out ways to mod things that are cheap to get expensive quality stuff), then this isn't the hobby for you!