I'm so sorry to hear about your situation!
So whether the dog is a "FEMA" dog or a search dog doesn't matter. By law, they can't enter your property.
Here is what you can tell the dog's owner:
This appears to be a trespass to land. Trespass to land occurs when someone enters onto the land of another. They do not have to intend to enter onto the land, they just have to be on the land of another without authorization. A person is not a trespasser if they were invited onto the land (e.g. houseguest) or if they are contractually obligated to be on the land (e.g. an employee at the corporate office). Otherwise, it’s a trespass. A trespass can even occur if a defendant causes someone or something to enter onto the land as well. The usual measure of damages are either nominal damages (one dollar) if there is no actual harm done to the land or actual damages if there is harm done (e.g. if the ground has been demolished by a vehicle on the land).
You may want to consider just writing a formal demand letter that they remove themselves or their items from the land. There’s a site that I’ve used in the past where you can find templates for various legal issues, including a demand letter (click the link - https://www.etsy.com/listing/674752957/demand-letter?ga_order=most_relevant&ga_search_type=all&ga_view_type=gallery&ga_search_query=demand+letter&ref=sr_gallery-1-1 Lawyers use letters like these often to enforce their client’s rights. It only costs $12 and it is about 90% cheaper than hiring an attorney to do it.