The thermal stability of metal carbonates depends on the polarizing power of the metal cation. A cation with high polarizing power distorts the large carbonate electron cloud ($\text{CO}_3^{2-}$), making it easier for the compound to decompose into a metal oxide and carbon dioxide on heating.

1. Alkaline Earth Metal Carbonates: As we move down Group 2 from $\text{Be}$ to $\text{Ca}$, the ionic size increases ($\text{Be}^{2+} < \text{Mg}^{2+} < \text{Ca}^{2+}$). This decreases the polarizing power of the cation, thereby increasing the thermal stability of the carbonate. Hence, the order is: $\text{BeCO}_3 < \text{MgCO}_3 < \text{CaCO}_3$.

2. Alkali vs. Alkaline Earth Metals: Alkali metal cations (like $\text{K}^+$) have a lower charge ($+1$) and larger ionic radius compared to alkaline earth metals. Thus, their polarizing power is much lower. As a result, alkali metal carbonates (except $\text{Li}_2\text{CO}_3$) are highly stable towards heat and generally do not decompose easily (they melt instead). Therefore, $\text{K}_2\text{CO}_3$ is the most thermally stable compound among the given choices.

The correct increasing order of thermal stability is: $\text{BeCO}_3 < \text{MgCO}_3 < \text{CaCO}_3 < \text{K}_2\text{CO}_3$.