At any time before proceedings are begun for such appraisement, or before action is brought for the recovery of damages, the owner or the owner’s agent may tender, to the person aggrieved by the depredation of such animal, the amount of damages which such owner may believe has been sustained. If the tender be accepted, no further damages shall be recovered in any way; if refused, and the person aggrieved fails to substantiate or recover as damages a sum greater than that tendered, no costs, disbursements, or expenses shall be collected or recovered in the aggrieved person’s favor, but the aggrieved person shall pay the costs and disbursements of such owner.