The PCM1808 is an audio sampler , optimized to run at sample rates of 44.1Khz for CD 48Khz for DAT etc. so the figures quoted are needed if you want to create an audio file that can be played by windoze etc .
if your app doesn't require that rate you can run it at any rate you like I quess , the important thing is the relationship between SCLK,BCLK,LRCLK and Dout ,as defined in the I2S standard
if the PIC doesn't have an I2S port it will be difficult to do in software
,for any sample rate fs , bclk = 64*fs ,sclk= 256fs , LRCLK =fs
if you use master mode the chip will from sclk generate LRCLK,BCLK,DOUT for you but the software will need to sample LRCLK drop , then bit shift 32bit in sync with bclk wait LRCLK rise etc , so at 16.384Mhz , you get fs= 64Khz , bclk= 4.096Mhz have fun in software with that , or use a divided clock out pin
or use an spi device
if you need audio rates you will need a chip with a PLL to generate sclk from your clock , I used chips like wm8805 for spdif output , but that can from 12Mhz generate 12.288Mhz etc