I have sleeved dozens of Big blocks. Tell the shop you want at least a .125 wall sleeve for a 4.300 bore. A common sleeve about seven inches long. I can't remember the stock no. (Geezeritis). This will finish out to the .040 that you have and have plenty of strength.
Tho most places "Short fill" .125 over Big blocks. I have raced and run them on the street with no filler and had no problems. If you have access to sonic checking, that would be the only positive way th check wall thickness.
Some blocks have core shift and some cylinders are rather thin between the cylinders. I have bored a few blocks to 4.440 for sleeve and still had around .100 on the thrust side of cylinder before install of the sleeve.
I am doing a marine block for myself at this time. Got the block cheap on Ebay, local pickup and just needed a sleeve. Walls were thin on this block. So to insure that I had good strength. I ordered and installed a Darton .250 wall sleeve. Thats a 4.755 OD. Needless to say VERY LITTLE original wall left on thrust sides (.060 maybe) and none between cylinders.
This is going to be in a .060 over with 4.375 stroke. Blown, hot street motor for my 47 Stude P'up.