Hmmm...the relationship seems to provide both ContactId and FullName for me.
There's a slight chance you're just missing the control field selector...
I created "Demo Project Data" entity.
Created Lookup relationship it to the Contact entity
Created a new app off the "Demo Project Data" entity
On the EditScreen screen, with the EditForm card selected, there is by default a Contact Dropdown.
If you select the Contact card or its dropdown, the properties screen on the right shows the cards, you can click the "..." menu to access configurable options as well as advanced options for a specific card. For the Contact card/dropdown;
The default Display Fields entry is 1.
Below this you have two fields you can edit. By default there is only 1 field displayed, so the value of the first field will be what you see in the dropdown - which is ContactId by default.
If you click the dropdown arrow next to this, you can select FullName (so both fields 1 and 2 will be FullName....you're only displaying Field 1, so it doesn't matter if they're both the same)
I just changed the first lookup field to be FullName.
Word of caution though - as I mentioned before, by default the "FullName" column for the Contact entity is EMPTY, so unless you've put data in there, you won't see anything....
Red circle marks the dropdown to select FullName as the display field