But several scientists have been referred to by their profession, rather than by their obvious commissioned rank. Say, Senior Geologist D'Amato, with Lieutenant's stripes, is never called "Lieutenant" by his colleagues - only Losira calls him that. And some department heads such as Chief Engineer or Chief of Security don't have the word "officer" attached to their titles, nor do they necessarily get referred to by their rank if their title is more informative in the context. Spock seldom gets called "Commander", either - but his title features the telltale word "officer", unlikely to fit enlisted status (even though I guess that could also be possible in certain special cases).
I'd thus see enough leeway to let Tomlinson be both a Lieutenant and a Specialist at the same time...
I can buy an officer being referred to by his actual job description, such as Geologist or Historian or whatever. But "Specialist" is obviously not a specific job description, and it is definitely an enlisted rating in a naval service -- particularly when you consider that Angela Martine is listed in the Concordance as a Specialist 2/C, i.e. a Specialist Second Class. That's a rating, not a job. So saying that someone can be a lieutenant and a specialist at the same time is like saying he could be a commander and a chief petty officer at the same time.