The Fermi-Dirac function has a very long tail, so will tend to occupy states a long way above the Fermi level. This can mean that you need to include more conduction states in your calculation, in order to capture all of the occupied states, and have a well-behaved optimisation method. This is probably why your calculation is not converging in this case.
Different codes should not need bigger smearing widths, if all other parameters remain the same. The only slight issue is the definition of smearing width; most codes use the standard deviation of the distribution, but occasionally you will see the full-width half-maximum used instead (and similar alternatives). It is straightforward to calculate (or just look up) the relationship between the two, if you do need to convert the numbers.
I disagree with the statement "The natural way is Fermi-Dirac and 1500 K is way too big". This appears to stem from a misunderstanding of what you are trying to accomplish with smearing. The goal of smearing is not (usually) to compute the thermal occupation of the Kohn-Sham states, but to: (a) improve the numerical approximation of the integral of the band-energies and occupancies over the Brillouin zone; and (b) improve the conditioning of the Fermi-level search. This is why you can use smaller smearing widths if you increase the k-point sampling density.
In the case of (a), the "natural" smearing is actually Gaussian, in the sense that it is related to both an uncertainty estimate and using Gaussian quadrature to compute the integral, rather than the raw numbers. For both (a) and (b) you want the width of the smearing function to be large enough to provide a smooth variation of the energies across the Brillouin zone; for typical k-point sampling densities, this means 0.1-0.2 eV.
Using a smearing function does change the calculated ground state properties. For the ground state energy, the first-order error can be removed for all the usual smearing schemes, but in the case of Methfessel-Paxton and "cold smearing" you can do better than that, and remove the 2nd order error and even higher (depending on the order of the scheme). The smearing does cause a stress on the cell, essentially thermal expansion, and I am not aware of anyone having calculated a correction for this.