Joist pulling away from center beam

It's not uncommon to create a ledger by laminating dimensional lumber, as you seem to propose. You'll see this for instance with deck building.

When using a through-bolt (through the 6in width of the beam, nutted on the far side) you have to pre-drill to the diameter of the thread, and so there will be no splitting. Make sure you use washers on both ends of the bolt.

Since the joists will no longer be seated you have to use hangers, as you correctly propose already.

Is this beam and are these joists floor or ceiling?

Whichever it sounds risky to cut them off without considering it at length. That the joists have not failed at the ends in 140 years points to leaving the ends alone. Installing a new ledger held to the beam with bolts would depend on the bolts and friction to carry the joists. Of course, maybe with modern construction adhesive. . .(edit) between the ledger and the beam the stress would be taken off the bolts.

If I understand the picures, right now the top portion of the ends of the joists are fairly tight to the beam. It seems that you should figure out a way to insert hangers to secure the ends of the joists directly to the beam.

Perhaps you could raise the joists slightly to allow room to insert hangers. According to you the load is carried by this 2x4 ledger which has sagged. Are the joist ends nailed to the beam at the tops?

Perhaps it would be sufficient to use a pair of angle brackets on each joist, one on each side. But maybe it would be better to figure out a way to insert a hanger around each joist.