Does Relentless Pursuit allow me to move an unready unit to a battlefield?

Yes, Relentless Pursuit allows you to move an exhausted (unready) unit to a battlefield.

According to RiftJudge FAQ #10072 and RiftJudge FAQ #8882, when a unit is moved by a card effect (such as Relentless Pursuit), it does not exhaust as part of the movement, and there is no requirement for the unit to be ready to be moved. Since Relentless Pursuit simply instructs you to "Move a friendly unit" without mentioning its state, an exhausted unit can be chosen and moved, and it will arrive at its destination still exhausted.

This is distinct from a "Standard Move" (Rule 144), which does require exhausting a unit as a cost. Because Relentless Pursuit moves the unit via a card effect, it bypasses that cost entirely.