You appear to have these relationships:
People::_fkClientID = Clients::__pkClientID
Invoices::_fkClientID = Clients::__pkClientID
Though your field names are likely different from mine.
In People, define a text field, FullName, with an auto-entered calculation such as this:
LastName & ", " & FirstName
Set a unique values validation option on this new field.
Then go to Manage | Database | Relationships and use the duplicate button (two green plus signs) to create a new table occurrence of People. Link it to Clients like this:
Clients::_fkPeopleID = People 2::__pkPeopleID
Format _fkPeopleID as a drop down list or pop up menu using a value list where field 1 shows data from People::__pkPeopleID and field 2 shows data from People::FullName. You can hide the first field to list only names if you want.
Now you can put People 2::FullName, People 2::firstName or People 2::lastName on any Client or Invoice layout that you want and you'll see the name of the designated "invoice person" on that layout.
For an explanation of the notation that I am using, see the first post of: Common Forum Relationship and Field Notations Explained
Your instructions were clear, I understood them perfectly and it's exactly the solution I needed.
I got a strange result at first, everything linked correctly but I couldn't hide the personID field in the dropdown list - if I tried to do so, or even to simply sort by the second field (personFullName), I got a question mark. I found the error was that the FullName field calculation had been left as "number" and I needed to change it to "text".