First a question in regards of the weapon customization. Does the speciality gear count as standard kit for the purpose of customization? I am not sure, but thought not.
Well now for the real topic. One of my player wondered why nobody, not even the Tech-Priest Enginseer can customize other weapons than that of the standard kit? Or a character with the right lore and tech skills? I mean I see the logic that a normal guardsman can not customize every weapon, but Tech-Priests?

It is a weird rule. I imagine the intent is to make it so that only "truly personal gear" can actually be customized. I would say there needs to be a mechanic to alter standard kit (e.g. the entire regiment is switching to hellguns).

As far as specialist gear, I would call it their standard kit.

I think its because all equipment you get via Logistics would have to be handed back to the Munitorum.

Well could be the case, but you can modify your regiment favored weapons, which by that logic would also be send back to the Munitorum one day, if you do not have it in your standard regiment kit. Is there even a rule that you must give something back that you get via a logistic test? Also a guardsman can raise his chance on the logistic test via trading and commerce so I think that also covers the unorthodox way via barter with other regiments or other illegal means.

And if you acquired that autopistol "illegally," why couldn't you mod it.

The sense I get from the rules is that its intended to prevent guardsmen from modifying plasma/melta weapons to somehow make them overly effective.

Honestly, the game needs a way of allowing gear to become standard kit. It some sort of broad "the regiment has petitioned the Munitorium for a sidearm, and they have permitted it, everyone now has a laspistol." Effectively, some means of getting more supply points like the 30 we have at regiment creation. Possible mission reward? An option for permanently spending the squads Logistic rating? This would allow the GM to give thumbs down for "improper" choices (e.g. we want to drop 50 logistics to give the regiment power armour, or we want meltaguns to be standard kit), but provides an avenue for adding quite a few weapons/armour options.

Honestly, I'd say its for your GM to decide. Customizations can be pretty effective, quite a few of them give no meaningful negative (home materials, sacred inscriptions, custom grip most of the time, modified stock). The test itself is quite easy, and can actually be left to a single member of the party.

My question is: what happens if you try to get a better weapon, not in the "upgrade from las to plas" sense but in a meh-good-best way? Let's use the starting gun: Good does nothing for me as both options are reliable, Best is pretty nice but expensive. This creates a dilemma of upgrading to best, which is a different gun, or keeping your customizations. This is honestly no dilemma at all as the bonuses provided by customization are far superior to best quality in many ways and are much cheaper.

They need to make a way a newly requisitioned weapon might be made into a permanent piece of standard kit and then allow it to be customized.

To actually answer a question: I'm positive that the base gear from your specialization counts as standard kit. I'll support this with the Ogryn, he can only use the gun provided by his specialty, not his regimental kit, and it seems ridiculous that they would ask you to tart up a gun which you randomly destroy.