Big fan of critical role here, and am familiar with Matt Mercer's house rule regarding Bonus Action spells- which is what your GM may be confused about.
Now, as other answers (THiebert's, Purple Monkey's) have pointed out, casting at a higher level doesn't affect the casting time required.
Before going to the houserule, let's go over to what RAW says about casting two spells during your turn. The rule is tricky to find as it's hidden away in page 202 of the PHB in the last sentence under "Casting Time" - "Bonus Action":
... You must use a Bonus Action to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast a another spell during the same turn, except for a cantrip with a casting time of 1 action.
the emphasis is mine, take note of this, we will get back to it later.
This is a major balancing factor in the game to reign in spellcasters from flinging spells left and right. In the early months of Critical Role, Matt Mercer removed the restriction that only cantrips can be cast (see emphasized portion), and allowed any-level spells to be cast along with any-level bonus action spells.
This house rule changed, though, at around the time Orion Acaba (guy who played Tiberius, the Dragonborn Sorcerer) left the game (I don't have enough fan trivia knowledge to point out the exact episode number of the house rule change). Matt Mercer put a restriction that only spells of 2nd-level or lower or cantrips can be cast as an Action after the bonus action spell (or vice versa).
Effectively making the houserule like this:
Matt Mercer's Bonus Action (spell)
... You must use a Bonus Action to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast a another spell during the same turn, except for a spell of 2nd-level or lower or a cantrip with a casting time of 1 action.
The point of confusion, I presume, for your DM is that during their (Critical Role) games, the players sometimes argue whether or not they can cast this spell as a Bonus Action and then comes some clarification about the level of the spell slot being used, then someone (usually Matt) clears it up that because it's 3rd-level, you can't cast it as a Bonus Action. Rest assured, even if you cast Healing Word at 9th-level, the spell still takes a Bonus Action to cast. If your DM allows Matt Mercer's House-rule, you can also then cast a 2nd-level spell or lower as an Action (or vice versa)