ruffdove's answer is likely correct that, within the episode's scope, the bar's name is a reference to San Junipero's company's name. But there is more going on here than just that. "Tucker" is a commonly used name in the Black Mirror universe.
- Playtest - TCKR Systems is mentioned in an article about putting nostalgia in a game.
- San Junipero - TCKR Systems runs the San Junipero project.
- Metalhead - Logo seen on a van
- Black Museum - Most of the stories revolve around TCKR Systems' technologies.
- Bandersnatch - The company is called Tuckersoft
- Striking Vipers - TCKR Systems developed the VR technology for the game being played.
Wiki with references to TCKR in many episodes. Second link with more.
While Black Mirror is obviously light on the continuity, since it's an anthology series, there's an implied thread here:
- Company Tuckersoft, in the 1980s, created Bandersnatch
- Company Callister Inc, in the future, makes a VR-style game using the technology seen in San Junipero. It is revealed that this technology is new and unfamiliar to people
- Company TCKR Systems, in the further future, is the company behind San Junipero and similar VR-device experiences. Here, people are commonly aware of the technology.
Given that Callister Inc is creating a video game (of sorts) like Tuckersoft did, and lies at the basis of the VR technology that TCKR Systems later uses. There seems to be a clear progressions here, where the development of games leads to VR technology which in turn leads to non-gaming-related usages of the VR tech.
I'm surprised Callister Inc was not named after something "Tuckery", though since this is the advent of the "white knob" VR technology, it may not have been planned to be such a recurring feature initially.
According to this Reddit comment, "Tucker" is a censoring of the word "Fucker", which Charlie Brooker (the creator of Black Mirror) wanted to name companies in the Black Mirror universe, but wasn't allowed for obvious reasons.
I cannot find further corroboration of this claim, but I have seen several appearances of Charlie Brooker in interviews etc. and I would say it fits with his general attitude. I consider this plausible but not proven.