Will a 4x4 be suitable?

I would probably go with a 6x6 and notch out two of the faces, without doing any math. Takes the weight from the bolts and allows them to provide clamping pressure only. Hard to say for sure without knowing exactly what your doing.

You've got hydraulic jacks - even cheap ones will lift 2", so lift height isn't critical, so don't try to put your 4x4 vertically - that could be wobbly as it moves during the lift, and it concentrates all the lift force at one point on your structure.

• Lay 2 4x4s down sideways so that they cross all the 2x12s near each end.
  • Leave clearance for the existing supports and the future supports
• Use hurricane ties or joist hangars (with appropriate nails or screws) to attach the 4x4 to each 2x12
  • This is to prevent movement, not to resist lifting force.
  • All the weight is transferred directly wood-to-wood and you're not counting on any piece of hardware resisting the sheer force of the weight above.
• Put one jack at each end of the 4x4, 1 or 2 joists in from the edge and directly under a joist
  • This will spread the force across all the 2x12s while putting the most direct force directly under a joist.
• Do the same at the other end of the joists.
• Enlist some friends/family to operate all 4 jacks at the same time and same speed to ensure all lifting happens evenly to prevent racking of the structure.
• Jack it slightly higher than the final height.
• Set up the new footers/foundation/final supports
• Lower onto the new supports and bolt/lag/nail into place.

This will give you 6 tons of weight plus the weight of the structure itself spread across 4 jacks. Theoretically, each jack is only taking about 1.5 tons. If you get four 6-ton jacks (which you can pick up pretty cheaply at the store that goes to the Harbor to get their Freight, or any automotive store), you'll have more than enough jacking capacity.

If the jacks at full extension don't reach the height you need, support each jack on some 2x6(or greater)x12" cuts stacked (and nailed/screwed to prevent movement) until the compressed jacks just touch the bottom of the 4x4.

I used a system similar to this to hold up our porch roof (on a 6x6) when we tore out the porch & columns below it to rebuild them. Supported some 2x10 on scaffolding, put the 6x6 on top of that, worked the jacks under, lifted the roof, put 6x6 support posts under it, released the jacks, removed the scaffold. Worked a treat. Used jacks from the aforementioned store with no problem - for the few extra bucks, I got two 12-ton jacks - significant overkill, I'm sure, but they held with no worries.