The big problem with the horizontal portion of the wedge is the thickness & have an angle on the underside of the horizontal portion of the wedge. By having only one edge beveled & on the top side of the horizontal splitting edge, you will not have the lower split piece of wood being wedged between the beam & the wedge. Get a piece of 1/2" thick plate 16" long & 6 - 12" wide. Along the 16" edge cut/grind a bevel on one side only at approximately a 35-40 degree angle. When you weld this to the top of the vertical wedge keep the beveled edge up & positioned around 2" back from the vertical wedge. This way you will be splitting in 2 stages, left/right, then up/down. I would also use 1/2" flat to make a vertical wedge to go on top of the horizontal portion which is 8" tall. Bevel both sides of this piece & have it lined up with the bottom edge.
It is possible to make this add on "4-way" removable. It is not an adjustable height, but it is easy to add on & remove. In a nut shell I had a piece of 2.5" square tubing welded to the back of my stationary wedge . I slid a length of 2"square solid bar down into it then added another piece of 2.5" tubing to keep it uniform. This is what supports the upside down "T"
The upside down "T": The horizontal edge had a "V" notch cut into it to which I welded an 8" long piece of 3"x3"x1/4" thick angle iron. The upper vertical splitting edge was welded to it & the horizontal flat creating the upside down "T". The angle iron is then butted up to the 2.5" length of tubing which stick up above the fixed wedge which gives it the 4-way/cross look. Then cut a couple 1/2" lengths of the angle iron to go on the backside of the 2.5" tubing. Weld one to the angle iron on the wedge towards the bottom & one at the top. You will have to shorten these small pieces of angle iron so you have a close to the tubing but not a tight fit. I used angle iron instead of 3" tubing on the "T" so if the 2.5" wedge mount should bend any and the 4-way option were to get stuck, you only have to cut a half of inch of angle iron on the bottom retainer, spread it a little & off it will come.

To keep the vertical edges in alignment, I welded 2 small pieces of steel to the underside of the "T" which cradles the lower vertical/fixed wedge. (See 2nd picture)
I have split several cords of wood with this wedge system & am very happy with it. A couple of additional benefits: 1 - The taller wedge cuts the "strings" allowing the rounds to fall apart more consistently. 2 - You can stack up 2-3 smaller rounds & split them at the same time.