I'm talking about a lawyer who would defend a urdp case that he or she sees is totally baseless.
I've done cases for cheap or free in very limited situations. I've also taken cases on a "I'll pay you what I can, when I can" basis from people I never heard from again after winning the case. So call me jaded, but I like to know up front if I'm going to be stiffed.
But, you are really missing the dynamics here. In general, there is very little contingent fee litigation in business matters. The proposition here is that someone should spend their time for free, so that you can make money as opposed to "get on with your life after a debilitating injury" in situations where damages are readily calculable in terms of medical expenses, lost wages, continued disability and pain, etc.. Personal injury litigation works on this basis because you have (a) someone who was demonstrably injured, (b) someone who is clearly at fault, and (c) all relevant events occurring in the same jurisdiction.
Let's take a typical situation. Say you are located in Mexico, your domain name is registered with a registrar in Australia, and you have a UDRP complaint from someone in Europe. And let's say it is a totally ridiculous case which you hire me to defend and win on this "you'll get paid after you do the UDRP, and then file a lawsuit somewhere" basis. I'm a US lawyer, and the UDRP didn't have anything to do with anyone in the US, so obviously I'm not going to be able to file a suit in Mexico, Australia or Europe without hiring another local attorney in one of those places - and that attorney is going to want to get paid.
Or let's take a simpler situation - you are in the US, your registrar is in the US, and your complainant is in Europe. I take the UDRP and win it. That doesn't somehow give you an automatic right to sue the complainant for much of anything in the US in all instances, and in the situations where that has been tried, the first thing the complainant does is challenge jurisdiction of the US court, because the UDRP admission to "Mutual Jurisdiction" is worded to include an admission to jurisdiction in the event the losing respondent wants to contest the outcome.
Can you sue a UDRP complainant for something? Yes, sometimes. I've represented, and currently represent parties doing precisely that. But only sometimes and only in the right set of circumstances. There have also been situations where after a reverse hi-jacking decision, the threat of a lawsuit has been enough to obtain a settlement from the complainant. Those are, in general, confidential, and so it is something that you and others never hear about.
In the absolute simplest type of situation - a US domain registrant, a US registrar and a US trademark claimant - the first thing that is going to happen is that the complainant, along with their reply to the suit, is going to include counterclaims against you to the tune of $100,000 under the US cybersquatting law. Their claim on that point might not be the best, but you are going to also have to defend the counterclaim in the course of proceeding with your claim.
Again, contrasting that with the car accident situation, the guy who ran the red light and hit your car doesn't have anything to sue you for - it's purely a matter of liability to a calculable amount having been established, and negotiating the compensation due from the other guy's insurance company, which solely cares about the numbers, under the threat of taking it to court.
One ride I've been taken on, and won't do again, is a UDRP defense on the proposition that the complainant will want the name, and that my fee would be taken from whatever might be negotiated after winning the UDRP. Done that several times. Won't do it again. Each time, except once, after winning the UDRP, the clients decided their names were worth utterly unrealistic figures. I can't, as a matter of principle, decide for other people what they want - only they can do that. In the other instance, the client went ahead and sold the name to the complainant later on anyway. That person is located in another country, and if you know a lawyer in that country who takes breach of contract cases on contingency, then let me know, because the cost of going after someone halfway around the world for a couple of thousand dollars is more than the value of the recovery.
Same thing with the "I'll pay half now and the other half if you win" type arrangements. That simply translates to "I'm going to stiff you for the other half and there's not much you can do about it."