The only downfall of the 338 Spectre is the range. While it can be shot further, it's really meant for close up work. Bullet selection is also critical, and the specialized bullets the Spectre needs to be effective (at least by my standards) may not be available to the OP. I am shooting monos designed specifically for this caliber by Maker Bullets. 300gr subs built to expand at Supersonic performance could certainly be increased by utilizing a 16" barrel instead of the 10.5"338 Spectre in an AR-15 would be a neat option.
In the EU, a metric caliber might be a bit easier to work with but, a 375 Socom or similar would be a good option.
The other options are pseudo wildcats like the 8x57 and 30-06 variants though subsonic use probably rules those out.
A 9.3x57 would not work well as a subsonic option but, that or a 9.3x62 would put a world of hurt on a large Russian Boar.