Intuition of Shephard's Lemma in Reverse
Expenditures on good $2$ are, indeed, included and can vary.
We know from Shephard's lemma that whenever the marginal change in expenditure for good $1$
with respect to its price varies with the price of good $1$, the Hicksian demand for good $1$ must vary too. But since utility is fixed, changes in the Hicksian demand for good $1$ require changes in the Hicksian demand for good $2$, a move along the indifference curve. And since we do not change the price of good $2$, the expenditure on good $2$ must vary too.