In short, it is extremely challenging if not impossible to detect SNe Ia before we see them.
As you said, the neutrino signal would be even weaker than for core-collapse events, and even that is pretty weak. We only caught one or two dozen neutrinos from SN 1987A, and that essentially went off in our own galaxy.
No other particles would escape faster than light could diffuse out of the explosion, even if they could be accelerated to ultra-relativistic speeds.
The only other long-range signal would be gravitational waves. The problem is that if SNe Ia are triggered by slow mass accretion onto a white dwarf (the single degenerate channel), the quadrupole moment will be no stronger than for any other binary star system in the universe. Even if white dwarf mergers turn out to be responsible (the double degenerate channel), some speculate that they could be third-body-induced head-on collisions rather than inspirals, so again no quadrupole moment to speak of.
There is a chance most SNe Ia are the result of inspiralling white dwarfs, which is the best case for emitting gravitational radiation. In this case, though, the signal is even weaker than neutron-star/neutron-star mergers (which are in turn weaker than the black-hole/black-hole merger so far detected), simply because the masses are smaller. On top of that, the inspiral (and gravitational wave emission) ends shortly after the surfaces collide, which is well before the objects' Schwarzschild radii merge, so the chirp in the signal is cutoff before its strongest point.
Also, unlike in the core-collapse case, there is hardly anything worth looking at just before the explosion. White dwarfs are hard enough to spot nearby in our own galaxy; they are next to impossible to see in other galaxies.
On the bright side, SNe Ia actually grow brighter for the first week or two after explosion, so there is still a sense in which we can catch them early. Indeed this and the next generation of high-cadence optical surveys are making this more and more common, as there is important information to be learned from the rise of the light curve.