“I know of no Calvinists who claim that God only cares about Calvinists.”
Really? I know that the Calvinists all believe that God loves them because He pulled their names out of the Bingo wheel, but what about the bulk of humanity?
The WCF says that:
“7. The rest of mankind God was pleased, according to the unsearchable counsel of his own will, whereby he extendeth or withholdeth mercy, as he pleaseth, for the glory of his sovereign power over his creatures, to pass by; and to ordain them to dishonor and wrath for their sin, to the praise of his glorious justice.”
So God loves those that he sends to everlasting hellfire?
As Frumanchu said, God loves His elect, His family whom He named and numbered from before the foundation of the world, who are found in every nation and race on earth and reside within all churches where the word of God is preached in truth and light (some more than others.)
The conclusion you're defending is a non-sequitor. It may be properly said that most Calvinists believe God only loves the elect, but given the fact that (with the exception of some "hyper-calvinists") we don't believe only Calvinists are elect, it pretty much renders the conclusion you're defending a fallacy.