In this case, the wavefunction collapses to one of two states: either the electron hits the screen and is absorbed, or the electron does not hit the screen and passes through the slits. In the latter case, the state it collapses to has peaks for both slits at once: it is the sum of the two wavefunctions that would arise if either just one or the other slit was open.
A wavefunction doesn't have to collapse to a single-point position state, it can collapse to a 'narrower' but still spread-out wavefunction.
However, it should be noted that measurement being synonymous with interaction is mainly a feature of no-collapse interpretations, like the Everett (Many Worlds) Interpretation. Collapse interpretations usually make a distinction: to trigger collapse and an observation/measurement, the observer has to be 'big enough' in some sense that it acts classically rather than as a quantum system; or, in some interpretations, the observer has to be 'intelligent' or 'conscious'.
The screen with the slits in it is certainly big enough to pass for those interpretations relying on mass or number of states to trigger collapse. It's obviously not conscious/intelligent, so it would not pass the 'Schrodinger's Cat' test in which we suppose that the cat doesn't collapse the wavefunction; we need a human to open the box to do so. If a cat can't do it, an inanimate screen certainly cannot.
For the no-collapse interpretations, we have to understand that the appearance of collapse is observer-relative. From the point of view of the system interacting with the electron (like the cat in the box), the wavefunction appears to have collapsed, but to an external human observer who has not yet opened the box, it has not. This is because when systems interact, they enter a superposition of non-interacting, orthogonal states, each of them consisting of an observer seeing one outcome. The cat in the box is a superposition of a cat seeing the electron spin 'up' and a cat seeing the electron spin 'down'. Each part of the cat wavefunction thinks it is the whole, and the electron has collapsed to one definite state. To the human outside the box, both electron and cat are still jointly in a superposition, yet to collapse.
So as a human observing the double-slit experiment, we can ignore the slits observing/collapsing the wavefunction, because they only do so from the slits' point of view. Each version of the slits sees one outcome: either the electron hit the screen, or it passed through. From our external human point of view, the collapse has not occurred yet; it only (apparently) does so when the human observer sees the result. (And the human becomes a quantum superposition of mutually-invisible observers each seeing the electron in only one place.)
In summary:
In this particular case, the collapse is to a 'through both slits' wavefunction which gives the expected interference pattern, and so isn't a problem. We're not ignoring it; it's actually required.
Collapse interpretations allow interactions not to trigger measurement/collapse if the system is not conscious, or very small/light. The answer depends on which interpretation you're using.
No-collapse interpretations (e.g., Everett/Many Worlds) are the ones for which interaction is generally synonymous with measurement/collapse. But in these the appearance of 'collapse' is specific to the observer doing the interacting, and other external observers don't see it until they interact, and can ignore it.
