If your rear axle is stock the Custom Constellation would be a 5567 and would have a 3.23 ratio if equipped with Ultramatic. There were optional ratios though so you may need to work out the ratio by counting turns if you don't find a stamp or tag on the center housing flange to identify which is in the car. The stamp or tag would be in a format 42-13 for the 3.23. If stamped it could be covered with dirt and grease.
Bendtsen makes an adapter kit for converting to the 700R4 type GM trans.
https://transmissionadapters.com/collections/packard/products/1955-1956-packard-352-400-v8-to-700r4 I believe a couple of other GM trans will also work the the adapter. Read thru Stewarts 55 400 project blog to get some idea of what is involved in the conversion and you may need to do some extra work on your own-- possibly fabricating the crossmember and throttle cable linkage since the kit apparently may no longer contain those items. The cable linkage may be the worst since welding to the original items may not be easy for some. There may be another option that does not require the fine detail welding and I believe it may have been briefly covered by Jappe who did a conversion on his car in the last year or so. The crossmember could possibly be handled by one of the modern universal units but don't know if anyone has tried. Trans work starts around page 46 in the blog. Work on your Clipper would be the same.
https://packardinfo.com/xoops/html/modules/newbb/viewtopic.php?topic_id=11305&viewmode=flat&order=ASC&type=&mode=0&start=450 Jappe covers some of the trans work on his Clipper here.
https://packardinfo.com/xoops/html/modules/newbb/viewtopic.php?topic_id=17504&viewmode=flat&order=ASC&type=&mode=0&start=0 Nothing needs to change with the TL.
As to the rear end, that usually does not need to change either but you will need some driveshaft work done to add a modern front U joint to work with the GM trans and maybe change the length. I would think most driveshaft shops could handle the work but if no one local wants to do it there is a place in Texas that has done work on Packard driveshafts with changing out the old ball and trunion joint to something modern.
http://www.driveshaftspecialist.com/Left-Column/Antique-Classics.html No idea what gears there are for the speedo but you might also need some kind of adapter box if a gear for the ratio and method of connecting the speedo cable directly to the trans is not available.
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