The Unicorn totem can't be triggered multiple times by the same spell.
The Unicorn Spirit is triggered when:
'[...] you cast a spell using a spell slot that restores hit points to any creature inside or outside the aura[...].
That's a three part trigger. To activate the bonus healing effect you need to:
- Cast a spell
- The spell you cast needs to use a spell slot.
- The spell needs to restore hit points to a creature.
Healing Spirit qualifies for these three triggers the first time it goes off - it's a levelled spell, that you've cast, that has healed someone.
However, the next time the same casting of Healing Spirit goes off it doesn't re-qualify. It's not a new spell you've cast and its not used another spell slot, so it's only hit one part of the necessary trigger.
Regardless of whether you find the above line of reasoning to be persuasive, the rules certainly offer no explicit statement that a single spell should be able to trigger the totem multiple times.
Further, many people already consider the Healing Spirit spell to be overpowered. Allowing it to proc the Unicorn totem multiple times via this, in my opinion tenuous, synergy in your game would only exacerbate that problem.
Alternatively, there's enough ambiguity here, that a DM could rule that Healing Spirit can't trigger the Unicorn totem at all
The totem is triggered when 'you cast a spell [...] that restores hit points'. Casting Healing Spirit doesn't restore hit points - not directly. The spell summons a spirit that the caster hopes will subsequently restore hit points.
So the casting of the spell does not restore hit points - though any subsequent restoration of hit points is a spell effect. Is that enough to trigger the Unicorn totem? I'd argue that its a bit quibbly to exclude it on these grounds - but it wouldn't be completely unreasonable.