SS advice needed...

From "welding engineering" viewpoint 316L TIG's really easily - so long as the weld looks right it is right. Can say that also from experience practicing / playing around. 304L is same for welding(?) - TIG it - looks right is right.
Matching filler. No reason not to TIG 304L with 316 filler.
The "L" is for (very-) low-carbon. These days thermodynamics has revealed a method of blowing down the carbon level in the melt in the steelworks so low there are no metallurgical complications welding stainlesses any more at all at all.
Rich S

304 is your best option.

I should have added that you can back-purge with Nitrogen or Argon, but never CO2.

I have both available. Since I don't have a dual argon regulator using nitrogen from my LN2 dewar would be convenient.

What do you actually get if you feed pure 100% nitrogen into a TIG torch? (if it works at all!) what weld do you get with austenitic like 304L? (difficult to confidently guess from science because lots of things would happen - eg. could end up with 100% austenite (no residual ferrite to mop us Phos and Sul) so get solification cracking like 310-series are supposed to be prone to...)
Different thing but...
Ar/2.5%N2 balances the nitrogen level when TIG'ing superduplexes (which have Nitrogen as a necessary alloying element) - neither gain nor loses Nitrogen when TIG'ing superduplex. Meet it on WPS's. Anyone comment about it in practice?
Ar/around 1%N2 recommended for (22%Cr) Duplex - which also "balances" Nitrogen - neither gain nor lose... Also meet on WPS's
There's Ar/5%H2 for austenitic stainless... Anyone comment on that? Tempted to get a bottle delivered to a welder Coding place I go to and have a play with it - see what it really does and get a feel for it...
Rich S

The torch still gets Argon, but since I only have a single Argon regulator, back purging with nitrogen from my Dewar will be convenient, requiring nothing but some hose from the 10 PSI gas outlet.
This is just a tamale extruder, not a space shuttle part so I think I'll be ok :)

Nitrogen is fine cold, but in an arc it becomes an active gas, and has very different properties.

I have yet to run into the Argon/Nitrogen mixes, other than reading about them.
The Argon/Hydrogen mixes have been around for a while for welding sanitary and marine stainless with less problems with chromium carbide precipitation.

So nice and cold from the dewar it should work fine for the back purge.

"cold" as in not in the arc!!! Nitrogen would become active pulverised by the arc. But as N2 gas, is very inert (?). Up to a few hundred Celsius?
Have I got that right?
Rich S