If you projected the file and then you saw that it failed to display, then ArcPad cannot fix that. If your uncompressed size of your .sid is 4.6 Gigs, than the projected Tif will be at least that size. Be sure that you have that much room to create a file like that and if you are using a network location, it will not time out. This time I would check the size of the file created and the display. If the output is bad again then could also be the version of ArcMap you are using. If you have ArcGIS 10, then I would think the Big Tiff, which I am not sure would display in ArcPad anyway. The best route may be to utilize a LizardTech license and then project it back to a sid in that software. ArcMap cannot export a sid to maintain the compression.
I would also consider sticking with the coordinate system of the sids and move the vector data, since the exporting and projecting of the sids can create more problems. I have also been advised to check out the vector data and then add the sids in as an SD card, otherwise the apm will be huge and cause problems.