I could also imagine that being forced to pay child support in a situation where I had been abused in this manner, and did not have custody of the child, could fuel a lot of very negative emotions in me.

Here's the thing though: you're describing a hypothetical situation that's on your side of the fence, your personal mental and emotional reality. But the rationale of the current system is the child as the claimant of both parents' support, regardless of the relationship between the adults.

The only consistent alternative model I can think of consists in a complete divorcing of the biological from the legal fatherthood - with the mother being the only presumed legal parent (with all rights and all responsibilities to the child), with fatherhood being an opt-in system, contingent upon the mother's agreement with the opt-in. This is NOT what the MRAs argue for - they would like the automatic rights to the child once born, on the same terms as women, but being able to back off the automatic responsibility for the child once born (a lower degree of control over conception in the first place, and no degree of post-conception and pre-birth control over what happens with the child, stems from biological realities and higher order principles that we're not disputing here). That's not a "ubi commoda, ibi et incommoda"-compatible, in presence of such a disparity of direct biological investment into the offspring. A consistent system would be an opt-in, rather than an opt-out (LPS): men would need to have no automatic rights to the child unless contracted with the mother, but there would be no automatic responsibility either.

The system can only work with two prerequisites. One, legal and actually accessible no-nonsense abortion (reasonable time frame, no waiting periods, ultrasounds, lawyers for fetuses or whatever newest idiocy they came up with) that's never on the political table (enshrined in law as a fully-fledged human right, full discretionary control over one's body and no obligation to allow its use to third parties). Two, a reworking of certain technicalities WRT child abandonment, as now women would have to be liable to such charges where they aren't, in order to ensure that all of these automatic rights to the child really are paired up with the automatic responsibility, to avoid a situation in which the taxpayers pick up the slack. Children could still be given up for adoption, but it would have to be accorded (pre- or post-birth) a bit differently, to ensure that the transfer of the rights & the responsibilities doesn't involve additional parties (i.e. the taxpayers).

Marital contracts could come with an option to secure automatic equal rights and responsibilities to any born children - but they couldn't abridge the woman's bodily autonomy before the children are born (abortion still remains exlusively her competence, her decision and her medical privacy, regardless of the marital status). That way both parties could secure themselves - the mother against the father backing off financially, the father against the mother not being willing to extend the rights to the child to him.

I'm skeptical about a widespread support for such a model, but it's the only consistent one that doesn't involve shifting the financial burden on the state, nor any further abridging of higher-order principles (i.e. bodily autonomy and medical privacy), nor not accounting for the disparity in the direct biological involvement. Everyone loses, but it's the closest alternative to something fair. Motherhood outside of a marital/contractual context would become a greater financial and legal burden than currently (abortion still sacrosanct, remember, and probably plenty of couples willing to contract an adoption), fathers would lose all automatic rights and literally have to acquire them from the mothers. I doubt many people would agree to such a solution, but it's a possibility, especially in a world where the legal practice has already started to divorce "social" from biological parenthood, i.e. established the primacy of the former in jurisprudence (which is why some non-biological fathers in the current system have parental obligations because the child has acquired the right to their support on "social bonds" or "legal identity" basis - not getting into those details now, though).

This is the TL;DR, the actual hypothetical model is a bit more worked out than what I presented here, but as you can see the changes are quite radical for everyone involved.