Requirements for pumice stone, compressive strength, and bulk density

Hello,
I have a 40 cm thick and 100-year-old load-bearing brick exterior wall on a shed with a 4 x 4 m large gate. I want to have this gate bricked up. Masons and consultants recommend pumice (because of the price), the mason recommends lightweight (because of weight / easier work), the consultant recommends heavy (because of stability / compressive strength) (I do not need thermal insulation here and the structural engineer says it (structurally) does not matter which stone I use). I am now leaning towards a pumice stone with a bulk density >= 0.7, and since I will only order the day after tomorrow, I look forward to comments on this.

If the thermal insulation didn't matter, I would simply take the cheapest one. The hut stood with a gate, it will also stand built with masonry, no matter which stone.

I have a 40 cm thick and 100 years old load-bearing brick exterior wall of a shed with a 4 x 4 m large gate. I want to have this gate bricked up.

I would not brick up a gate of this size without considering the future purpose of the space that will be accessed through it. By the way, you haven't said anything about the lintel yet: concrete, wood, brick (in which bond)... take a photo... and also nothing about the threshold.

The shed has four gates on three sides. On one side, a new construction is planned, so the gate will be bricked up. There was no threshold; now there is 4m x 40cm x 80cm of concrete. The lintel consists of two double-T beams with bricks between the beams.

I had already considered installing a door from the planned new building into the existing shed. However, this would be (in the gate area) either in a small storage room or in a bathroom, in both cases unfavorable in terms of both access and space.

To conclude this topic, here is the photo (ok, you probably meant a photo before the masonry, but I didn't want to take the stones out again now). The masonry was done with KLB Plan-HBL 4 12 DF, bulk density 0.9. 264 stones and 5 bags of mortar plus delivery for €905.