> Would it be possible to fool the triangulation from the device, by arbitrary (or intelligently) delaying the mobile radio signals?

Not without messing up your ability to make and receive calls. Cell towers use precise timing and power-level measurements in order to do things like decide which cell-site is best, and to hand-over your call from one tower to the next without breaking your call or glitching.

Edit: Even if you were to play around with timing of responses of the radio signal, you have no control over how it radiates in free space. The time-delta between reception of the same signal by 3 towers at known locations is enough to triangulate your position. Maybe a unidirectional antenna pointing to just one tower might work, if there are no other towers within the beam behind it and no sideway leakages.

With highly directional antenna and carefully selecting your position, you could try to have your signal only to be heard by a single cell tower at the time. The network would get your distance from the tower, but with direction info from just one tower would be less accurate.

Expanding this, you could have N directional antennas pointed to N cell towers, and some individual delays on each of those antennas, it might be possible to fool the network triangulation. Such a setup would look highly suspicious if you were carrying it around, and it definitely wouldn't fit in your pocket.

There are no available cellphone radio baseband computers/transceivers that allow you do do things with that. You would literally have to implement the entire cell baseband from scratch with a software defined radio. It would be a very non-trivial project.

And it'd be useless unless you had many of these custom transmitters faking your signal spread out over large physical distances.

OsmocomBB and LimeSDR would like a word with you. Yes, the former is limited to GSM, the latter doesn't come with a TX amp and you'll need to supply suitable mid-power RF (no cooling for passives, carefull cooling of actives) antenna circulator/filter/switch, if you want to use your new amp. The hardware should be under 2k$ manufacturing in single-unit quantities, but it is HF design, including some distributed-element filters and power-handling at low GHz frequencies. Nothing particularly trivial to design, though the requirements in precision are not too stringent, so you won't need someone who can demand >100$/h while working outside of a major metropolitan area.

TLDR: GSM+LTE open-source SDR/hacked dumbphone baseband exists, suitable hardware is COTS for sub $2k.