Frege's Got Zero Reference

An expression of the form ‘the F’ where ‘F’ is a predicate defining any range of the unique properties of a variable, should be read as ‘the unique x satisfying F’ and symbolized as the term ’(ix)(Fx)’. Frege considers the term ‘(ix)(Fx)’ to be synonymous with a proper name. In a logical expression of the form ‘a=(ix)(Fx)’, where the term ‘a’ is a constant interpreted as a proper name, Frege considers ‘a’ and ‘(ix)(Fx)’ to be not only interchangeable, but to be synonymous when translated as meaningful linguistic expressions. So consider the term ‘a’ to be interpreted as the proper name ‘Alan’ and the term ‘(ix)(Fx)’ to be interpreted as ‘the only French person in this room’. Under these conditions, the logical expression ‘a=(ix)(Fx)’ in English would be read as ‘Alan is the unique French person in this room’.
There are notable limitations to this framework. The variable ‘x’ and the constant ‘a’ in the expression ‘a=(ix)(Fx)’ must refer to objects in the domain of all existing objects. Furthermore, all existing objects that are discussed by competent speakers have a unique description such that, that object is secured by an expression of the form ‘the F’. But not all proper names secure a referent, some proper names are found wanting of a referent – they are bearerless or empty. That is to say, there is no bearer for that name (e.g. it does not refer to an object in the domain). In such a case Frege adopts the convention that the referent of the name will be the number zero, and sentences containing bearer-less names will not express a thought that has a truth-value. Given this, the conditions for determining truth-value for any sentence f containing a proper name, of either the form ‘a’ or ’(ix)(Fx)’, is as follows:
i. If ‘a’ or ‘(ix)(Fx)’ occur in a well formed sentence f then f is true or false, if and only if ‘a’ or ‘(ix)(Fx)’ secure a referent.
ii. If ‘a’ or ‘(ix)(Fx)’ occur in a well formed sentence f, and ‘a’ or ‘(ix)(Fx)’ do not secure a referent then f has no truth-value.
To see how odd the zero convention is, consider the following sentences:
1 Santa Claus is zero.
2 Unicorns are the sum of 1 and -1.
Both 1 and 2 are true by this convention. So Frege should reject the convention that sentences containing bearerless names lack a truth value. But he cannot adopt a convention that such sentences express a false thought, while maintaining that the referent of a bearerless name is zero. This is a terrible consequence and issues of direct reference, rigidity, pronouns, and indexicals aside – this single convention appears to cripple Frege’s semantic framework. But intuitively, we do assign a truth value to:
3 Santa Claus is the bum that lives at the North Pole, only working once a year.
4 Santa Claus is fat and jolly.
This said, is there a way to resolve this issue for Frege, while maintaining the sense-reference distinction?

Note 1.
For a copy of "On Sense and Reference" use jstor, but the Geach and Black translation is best. Avoid the version with the title translated "On Sense and Nominatum" (near the end when Frege discusses conversational implicature - the but and and distinction - the Nominatum version translates a sentence as claiming that but and and have no sense.) My adobe acrobat is failing to work currently, otherwise I'd post the stable URL from jstor.
Note 2.
The 'i' above is supposed to be the iota operator. Contra the russellian elimination of the iota operator, I try to avoid equating Frege and Russell.