You know I'm an Elise fan but you're pushing more for an opinion.
However I'm an honest man and you seemed an honest man too, so you will get an honest answer. How's that?
I'm more a fan of HD650 than HD600. So my answer is based on your getting a HD650.
If it's a HD650, get the La Figaro 339. Very few amps, whether ss or tube amps scale HD650 as much as LF339 does. Now Elise with HD650 or HD600 is very good too. Too good in fact but I just think that La Figaro 339 is exceptional when it comes to HD650. Seriously I am amazed that the LF339 isn't as well known as she should be and for a street price of $750 straight from the manufacturer, Yuking.
Certainly buy straight from Yuking and get the adapters for EF86 too. For tubes just get a pair of Mullard EF86 and RCA 6as7g. That's all you need. You can tube roll later.
Certainly get the upgrade 2013 onwards model. Silver or black is your choice. Also IMO, choose the 1st set of tubes, which is the clearer tubes. The 2nd set of tubes are warmer and might be a better match for your HD600.
As I said before, Elise and La figaro 339 are voiced differently. Deciding which I like better is very difficult. Today I might tell you I like LF339 better. Tomorrow I might tell you Elise is better. Now you know why I ask you to toss a coin.
If you are in Sydney, Australia, you can come for an audition.