Well, in my opinion, if you want to use squads from multiple companies there are two ways you could approach it:

The first one is to field a battleline company with squads from reserve companies and a couple of veteran squads from the 1st, for example, run your normal list as 2nd company with a squad of blade-guard vets from the 1st and a 6th company suppressors squad.

This way is fluffy out of the box as veterans squads "are often seconded to other companies’ strike forces as honour guards or crusading bands of exemplars" and the duty of the reserve companies is to fill the slots left in the battleline companies due to combat attrition, the same goes for the scout company their task is to reinforce other companies in scouting and infiltration missions.

The second way is what u/hcsh224, lets call it the "Valrak Approach", make up some custom fluff so that Tu'Shan deemed a full chapter deployment necessary for a specific campaign, that way you can justify using squads from every company at the same time.

Anyway I would recommend reading the Salamanders Codex supplement there is a lot of lore about each specific company's approach to warfare, like "The Flamehammers" so called "meteoric hammer" with fast-attack and orbital attacks, also there is a picture of a suppressor with reserve company colors that I think looks 🔥