My Arduino pro mini is 5v and I guess the LCD too. I don't currently have a 3.3v regulator to begin with, I only have mini dc-dc converters which can do the job.
Remember, if at 5v your project draws 50ma and you use a LDO, when powering from a fresh battery at 9v, you will consume 50ma from your battery all the way to it's death at around 6v.
(This only works if your DC-DC converter takes 9v in directly and operates between 6v&9v) If you use a mini DC-DC converted, or 'buck regulator' set to 5v, with you project consuming 50ma at 5v, when the battery is new feeding the regulator 9v, the current at the battery will be around (assuming 85% efficiency, check converter data sheet) 32ma giving which you might think gives you around 56% more battery life, but, at the battery voltage drops, the current will increase to 50ma, so, lets say you will get 25% more battery life.
Now, powering your project with 3.3v, if possible, say your project now draws 32ma. If you had a 3.3v LDO, the draw at 9v will also be 32ma giving you a true 56% more battery life. However, if you were now to use the DC-DC buck regulator, at 9v, your project will draw only 14ma. This is now a huge savings, we are talking a 350% increase in battery life, however, due to current going up as battery voltage drops plus alkaline batteries last much longer the lest power you consume, using the battery charts for 10ma, lets say around 30 hours for the 9v batteries, 250 hours for 6 AA batteries, 100 hours for 4AA batteries.
Remember, unlike linear regulators where current consumed is equal to current at the voltage input, regardless of the voltage, with a switching buck converter, current consumed at the voltage input keeps getting less as voltage increases * by loss in efficiency. This is why the 6x AA batteries at 9v perform so much better than 4x AA batteries at 6v.
Your best bet is to always measure current with a DVM to be sure. If you have a bench supply, you can confirm the efficiency of your switching DC-DC converter with different power supply voltages (simulating battery voltage drop). You may need a hefty cap on the power going into your regulator to smooth out your DVM current reading.
Since I don't know what your device needs to do, I cant recommend much more. I have made a 9v battery powered PIC project with a 2x16 character display in the past, and, I actually powered on and off the LCD character display and everything else from the PIC when not in use and got everything down to an average 2ma draw at the battery giving a respectable 300 hours life, or if I wen with 6x AA cells, I could have had 1400 hours life.