Upcasting does indeed bypass limited magic immunity
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. (PHB p.201)
The general rule is that a spell cast using a higher level spell slot is actually that level spell for that casting for all relevant effects. There is no base level and upcast level. A spell is the level it is cast at.
There is no indication in limited magic immunity that the ability is meant to block spells cast with a higher spell slot. Allowing it do do so would be granting it an ability that it does not have as written.
Compare that to the Globe of Invulnerability which says:
Any spell of 5th level or lower cast from outside the barrier can't affect creatures or Objects within it, even if the spell is cast using a higher level spell slot.
This item has a specific rule that overrides the general rule about spells cast with higher level slots. The rakshasa's ability does not.
Jeremy Crawford has answered a similar question about the wizard evocation ability overchannel and come to the same conclusion:
Q: Can I use Overchannel to cast 3rd level spell fireball using a
7th level slot? Seems to read that way.
A: It won't work, since that
fireball is 7th level. The level of a spell matches the level of the
slot used to cast it (PH, 201).
Since overchannel only works on up to 5th level spells, a 7th level fireball does not qualify since it is actually a 7th level spell when it is cast.