There are two? versions of .577 - 500
The first and less common is a necked down .577-3 1/8" or .577-3 1/4", which is a lengthened .577 with the same head dimensions as the .577 Snider and .577/450. None of these will go anywhere near a Martini action

The slightly more common, (in its day) is the ,577/500 No.2, a Westley Richards cartridge from the 1870s
The case is 2 and 13/16 inches long and probably the longest case that will go "round the bend" of a MH action
The head and rim are smaller than the .577/450s etc.

I have no idea where you would get reamers etc., probably in the US but here (UK) and Australia?
CH4D do dies for the latter cartridges and there are many moulds to choose from

Be aware that if you use the Greener GP shotgun action (a litttle bigger) that there are more than one version, varying in strength
If you PM The Double D (or DD) on this forum I believe he posts on BMF and is much more knowledgeable than me on Martinis he will give you good advice

Centre Cartridge is the .577/500 No 2 against the much shorter .577/450. The left case is .500 - 3"

.577/500 No. 2 From "The Handloaders Guide to Cartridge Conversions"

BB do brass for the .577/500

TH44