That is a Weatherby XXII, which was made by Anschütz and is a 64 barreled action in a weatherby stock. The OEM barrel was a dud, I could never get it shooting under 1/2" reliably, so I re-barreled it last year with a 1416 D HB take-off barrel from a trusted gunsmith. Ammo wise, I've tried nearly everything available from our various suppliers of match ammo. I'm just dabbling with the tuner on it while I wait for my custom to be finished, then the tuner gets bored out to fit the custom barrel (it just happened to fit on this rifle as-is). This isn't the rifle to be pursuing ultimate accuracy with, but I am pleased with some of the high level results I've been able to squeeze out of it so far without too much trouble. It otherwise serves as my Silhouette rifle where I'm not allowed to use a tuner on it.
What are the details/specs of the custom you're having built? What kind of average are you going to be expecting with your custom?
Some things I've heard others say is that the 64 action uses the sear as the bolt stop, which leads to issues over a loooonnnnng service life. Might not be an issue in the next 10 years or so... but past 30 years... who knows? Here's a thread with some comparative information http://www.rimfirecentral.com/forums/showthread.php?t=897202&highlight=64+bolt+stop
At the heart of it, majority of the accuracy is in the barrel, and the 64's are fitted with the same barrels as the 54's so accuracy potential is about equal between the two and subject to the normal variance observed in any factory made product. A 54 action, in and of itself, isn't going to make the rifle more accurate. It is more along the lines of: Do you like the trigger options available with the 54's over the 64's? Do you like the stocks that are only available on 54 rifles? Is durability over a very long service life important to you? Can you manage the extra weight? At the end of the day in rimfire, trying to get the best accuracy once past the 1/2" at 50 yards mark becomes very expensive for exponentially smaller gains, and one might as well go full custom rather than play the odds with a factory built rifle.
While I understand they're both "factory production" rifles, I wonder if the 54 rifle is built with less of a high quantity sacrificing quality mind set. Such as tighter tolerances in the action, chambering+fitting the barrel, and barrel selection. Similar to the way Shilen sells their barrels as match or select match. Does Anschutz hand pick certain qualities of a batch of barrels to go on their 54 rifles... no one would really know, but I wonder. What demands the 3-4x price tag if it's still a roll of the dice without a quantifiable measure(aside from service life) of being the better rifle over a 64. Paying for the legacy of Anschutz and 54 action?
Then there's full custom rifles... a new match 54 is knocking on the price tag of a full custom, where hand picking what gets used is part of the purchase. Still kind of playing lottery on the barrel, but at least tolerances can be more dialed in as per request.