This should be fine
In 5e, the creature types are merely tags that interact with other game features (MM, p. 6)
A monster's type speaks to its fundamental nature. Certain spells, magic items, class features, and other effects in the game interact in special ways with creatures of a particular type.
So by default, being a construct does not confer any immunities or other traits such as not needing to eat, drink or sleep, like often cited for individual monsters under the Construct Nature lore entry.
If you do not add those, any imbalances introduced would stem from interaction with other features such as spells like protection from evil that specifically only work or do not work on those types or on constructs.
I do not think that these effects are important enough to constitute a major imbalance. The most relevant here is more of a disadvantage, as you cannot use healing like cure wounds on constructs, but you can simply resummon the steed should it die. A reflavoring to construct should be fine.