Miscibility of Molten Metals
Questions like this are not prone to easy answers. But, lets look at what is known about the liquids, then at the various binary diagrams.

Lithium - Melting point 453K, density at melt is 0.512 g/cm$^{3}$, or 0.0737 mol/cm$^{3}$

Sodium - Melting point 371K, density at melt is 0.926 g/cm$^{3}$, or 0.0403 mol/cm$^{3}$

Potassium - Melting point 336K, density at melt is 0.828 g/cm$^{3}$, or 0.0211 mol/cm$^{3}$   

Calcium - Melting point 1112K, density at melt is 1.376 g/cm$^{3}$, or 0.0343 mol/cm$^{3}$

Clearly Li has by far the smallest atomic volume in the melt.

The Li-Na binary, as you point out, shows a miscibility gap, tilted towards the Li side, but peaking around 300C. The K-Li system is less well studied, but it also shows a miscibility gap which peaks at perhaps a similar value of 300C. 

In sharp contrast, the Ca-Li system exhibits an intermediate compound (Li$_{2}$Ca) and can be described with a 'virtually ideal' liquid phase (C.W. Bale and A.D. Pelton, Bulletin of Alloys Phase Diagrams 8(2) 125-127 (1987)).

Given that the molar density of liquid Ca lies between those of Na and K it is clear that you cannot ascribe the observed miscibility gaps solely to atomic radii. 

To further complicate matters, lets keep looking at more binary diagrams. The K-Na diagram shows an intermediate compound, KNa$_{2}$ and no miscibility gap.  KNa$_{2}$ has the same crystal structure as Li$_{2}$Ca (both are MgZn$_{2}$ prototype). The Ca-Na binary shows -- surprise! -- a miscibility gap towards the Na side, peaking at 1200K. So, Ca and Na, even with fairly similar atomic volumes, do not want to mix in the liquid, while Ca and Li are perfectly happy to.

Bottom line - atomic radii do not really mean that much when predicting binary phase diagrams.  Even with comparable radii, you would be surprised at the weirdness than can be found on phase diagrams. As one example, Ag-Cu (both fcc metals) do not form a continuous solid solution, and have a slightly positive enthalpy of mixing in the liquid (not enough to form a miscibility gap).  Yet, Au-Cu (Ag and Au have similar atomic radii) form a fcc continuous solid solution (with intermediate ordered fcc-based phases at lower temperatures), and has a strongly negative enthalpy of mixing in the liquid. 

Have fund exploring binary phase diagrams (and ternary, ...), but don't rely on simple heuristics.