Melting points of compounds?
First, in order to compare mp's you need to know the mp's. You have two choices: 1. run melting point measurements or 2. find literature written by someone who has run the experiments. Perhaps this is a translation problem from your native language? Perhaps you meant predict the ranking of mp's for those 4 compounds. 

You state that CaCO3 melts at 811 °C. As far as I know, that is incorrect. First, the thermodynamically stable polymorph is calcite, which according to wikipedia has a mp of 1339 °C but which decomposes at temperatures above ~840 °C (also according to Wikipedia!). I hope you see this is clearly contradictory. My first hand experience is that it does indeed decompose at around 820–840 °C, so I am unable to explain how they were able to measure its melting point 500 °C above that. 500 °C is an enormous gap, but perhaps they measured it under some enormous pressure (far from STP) - I really don't know. 

The other major problem I'm having is understanding what you (or your teacher) means by "counterbalance the effect of electropositive field of cations ... and electronegative field of an anion." It seems to me to be meaningless, at least I don't recognize this terminology in reference to the electrostatic forces usually invoked to account for ionic solids. BeSO4 is an obvious problem since Be is known to for its tendency to form covalent compounds. Indeed, according to Wikipedia, anhydrous Be sulfate has Be-S (covalent) bonding. While you could predict that the covalent bonding would reduce the mp, what you'd do with the polymeric nature of the solid is beyond me - I've no idea how you or your teacher can explain that, except by hand waving. 

This leaves the Ba and Sr salts. Simple logic of ion size would predict the order Ba &lt; Sr and ignoring the ridiculous assertion on Wikipedia that the Ba salt's melting point is 811 °C while at the same time (you did READ the article, right?) stating that that is the temperature at which it a "polymorphic transformation" occurs. A little further digging provides a safety data sheet which claims a mp of 1400 °C "close to its decomposition temperature", but such data should be suspect. 

Your teacher's choice of picking carbonate as the anion was a poor one, in my honest opinion. Because of the tendency of these salts to decompose, rather than melt, you need to be very careful where you obtain your data from. But perhaps that is the lesson your teacher hopes some of you will learn - to not just accept a number because it is readily available (without subjecting it to critical analysis (who, what, where, why, when, how - which in this case means how reliable is the reference? does it reference a primary (academic) source? if not, an authoritative secondary source? if not, then you should seek independent confirmation. Such confirmation is hard to find on the internet, since one lazy site steals from another until the noise drowns out the signal. 

I really don't know if density functional theory (DFT) quantum mechanical modeling by supercomputers is able to accurately predict the mp's of these material. My guess is that because of the change in crystal structure(s) that it remains beyond our current capabilities (for some things, calculating the value has become reasonably accurate – as far as I know (and I'm no expert) mp isn't one of the successes. If it was accurate it would be a third way to "compare" the mp's.