No you can not use a different cell tech, the charging circuit will not be compatible...
Have you looked on Ebay? You can get one for about $4 delivered...
Heck almost any store in the US that sells batteries has CR1620 or CR1632 on the shelf... The CRs are slightly different (not rated or chemistry designed for recharging, and probably won't take a charge but you will be able to still use it as a battery backup when the device is off, in a pinch) they will work for the immediate time if it's an emergency... Just swap it to a LIR when you get a chance... Don't keep the CR in there long or when the unit is plugged in, and best to check it to make sure it's not overheating or what not to avoid trouble, basically just use it as a temp fix if you really need to, or wait for the LIR to come in from China...
Worst case if you have to and there is space use a LIR2016 or LIR2032... A little larger...
FYI the first two numbers are the diameter in mm, the next two are the mAh... The higher the mAh the 'thicker' the battery is, for example a CR2016 is half as thick as a CR2032...
Have you checked any computer repair shops that might have them in stock for motherboard clock backups?