It sounds like you can wait a bit if it saves money.
Given that. I'd order a hd44780 display or two off ebay.
They are much cheaper than from Mouser.
Mouser also charges quite a bit for shipping.
You can get lcds off ebay for $3-$4 USD shipped.
For low cost components, I really like these guys:
You can get 74HC595s for $0.20 USD. Less than half of Mouser.
Great prices on things like resonators, crystals, caps, leds, diodes,
1307 timer chips, ds18b20 temp sensors, transistors like 2n7000, 2n3904, 2n3906, strip boards, IC sockets, etc....
For even as low as $10 you can load up on all kinds of really nice goodies.
Their shipping is really low too, especially for small quantity low $ orders.
Using a shift register on the hd44780 type display is really easy.
(I think using 74LS164s are slightly easier but they are not always as easy to find and sometimes cost more)
I recommend using this library if you are going to wire up the shift register yourself.
Here is some additional background:
An example of how to use a 74ls164 on a strip board:
Not that you can use 74HC595s instead of the 164 but obviously the strip board
wiring will be different.
But I wanted you to see how you can use strip board for stuff like this.
I'd change it a bit to allow the strip board to be plugged directly into the lcd
by using 16 rows on the left side of the components.
With the exception of a 16 pin female header, you can get
all the other other the other parts you need to make a 2 wire lcd backpack
for about $1 if you get all the parts from Tayda.
(You could solder the lcd directly to the backpack or just use wires)
Which brings the total cost of a 2 wire LCD including the LCD
to about $5 if you get the LCD off ebay.
$0.70 strip board (can make more than one backpack)
$0.15 40 pin header (can make more than one backpack)
$0.01 1/4 watt resistor (for shift register AND gate)
$0.01 diode (for shift register AND gate)
$0.01 1/4 watt resistor for backlight
$0.06 10k trim pot