Let me through more beans in the pot here.
First let me say, there are many good actions, custom or commercial available and some have been mentioned in previous posts.
The large Sako TRG is already chambered in the 338 Lapua. This means the bolt face will already work for you. The WHOLE gun wholesaled last year for around $600.00. You want an action and decent trigger (Not like Jewell), take the gun apart and sell the items you don't need. Now you have a repeating action that is not in excess of $1000.00.
1. Is this going to be a LR hunting gun? I think from your post, that's what you have in mind.
2. Does it have to be a 338 Lapua?
3. Have you considered a 338 RUM or a 338 Tomahawk? As I'm sure you know, this case can be fired on a straight Remington magnum action and it is VERY easy to get triggers, bases and most anything for Remington actions. A good action bedding job and part way up the barrel channel will hold the longer barrel just fine. You can buy a whole Remington ADL magnum for about $350.00 to $400.00. Again tear it apart and sell what you dont need.
The Sako or Remington are both repeater actions.
I have several custom actions and a whole bunch of Remington actioned rifles and can make a fairly good comparison here.
I have customs and Remingtons on Match guns and both on LR and Ultra LR guns too.
I went with the Hall G action for the 338/416 Rigby Imp because at the time, I felt that a large heavy duty action would be better in that size case and using the 300 gr bullet exclusivly. I also have a Large Hart action on a 30/378 Weatherby.
Not to mention BAT or Nesika Bay actions here, I think for what you want, and MUCH less expensive, the TRG Sako will fill the bill with the cartridge you want. Some are used at Williamsport on larger cases then the 338 Lapua.
Weatherby also makes a large bolt face action that will hold the 338 Lapua and larger and it is a repeater too. I have seen those (whole
for $800.00 and $900.00.
It would be interesting to see how well the 338 RUM would hold up using 300 Gr bullets and a 32 to 34" barrel. Remington says it will. I know it will work great with 250 Gr bullets.
If cost is not a problem, go with a custom but, you do have several options here that will work quite well.
You have to ask yourself, will the Custom action REALLY be that much better then the TRG Sako or Remington in the Hunting (I think) application you want?
I would probably opt to save the difference between a $1250.00 to $1900.00 custom action for a TGR and put the savings in real good LR optics.
These are just some of your options.
If it were me, I would go with the Sako TRG action with the 338 Lapua and if it were the 338 RUM I would go with a Remington.
Just my .02 Cents worth.