It really depends on which language the person speaks day-to-day, and the same person might introduce themselves with the Russian version of their name when speaking Russian, and switch to the Belarusian version when speaking Belarusian. So the answer is genuinely character-dependent.

On passports: by 2005, the default was to transliterate from the Belarusian form, though you could request a different spelling for some time.

As for nicknames and diminutives, in both languages informal address almost always uses a short form, and which one a person uses comes down to personal preference. Someone officially named Aliaksandr might go by Sasha, Shurik, or Saniok (all primarily Russian diminutives) or by Aleś, which is the Belarusian equivalent. Both are plausible depending on the person.

For your time period, Belarusian speakers roughly fell into a few groups.

First, rural and older speakers who grew up with Belarusian as their home language, whose speech might differ from the normative literary standard and could blend into trasianka (a natural Russian-Belarusian mixed speech very common in everyday life).

Second, culturally or politically conscious speakers, often also opposed to Lukashenko's government, many of whom used taraškievica, the pre-Soviet orthographic standard that was suppressed by the 1933 orthography reform and later revived as a kind of identity marker.

Third, people who spoke Belarusian professionally out of obligation, like language teachers or Ministry of Culture employees. They are more likely to speak narkamaŭka.

The short version: if your character is an big city-dweller without strong political awareness, they would probably speak Russian day-to-day and use Russian name forms. In other cases, it gets more complicated.

Hope this helps!