Foreign Key with a Constant
You said you didn't want to add an extra column to a number of tables that need to link back to Test1 in this way (ie on ThingA, 3).

How about adding one persisted computed column to TestA that shows the value ThingA if ThingB is 3, and null otherwise? 

Then your foreign key references the new column only, based on ThingA in the referencing table.

alter table Test1 add SpecialThingA as
    (case ThingB when 3 then ThingA else null end) persisted;


and

FOREIGN KEY (ThingA) REFERENCES Test1 (SpecialThingA)


In other words - one new column on Test1, rather than a new column on Test2 (to hold '3') and Test3, and ....