With some really quick googling, I've found that there are DBM kits for Accuracy International mags for the Sav 110 series.
Since the bolt fact of the Axis is the same as the 110, should not an AI mag work? And they're plenty available in the .308 class of cartridges (.243 & 7-08).
"Tube Stocks"
www.aihpa.com/Sponsor/Tikka_T3_Tube_Stock/T3.htm
http://www.6mmbr.com/gunweek074.html
Now for the Axis, because the action already has it's own "body", there only needs to be access for the ejection port, rear of the bolt, and the bold handle thru the "tube gun receiver + barrel shroud".
The grip, trigger guard, can be a machined "lower" similar to an AR style, with some sort of magwell for common AI mags.
Due to the integrated nature of the Axis action, I suspect this can all be constructed from an aluminium tube, some sort of "lower grip/guard/magwell, the bolted mounting block pictured elsewhere in this forum, and some sort of rear butt (tube, solid, folding, whatever).
And of course, powder-coated to your hearts desires.
But the real question *I* have, will the AI mag work? Not with the stock Axis stock. But with the bolt and action in general.