What is the role of a CFI (not CFII) as safety pilot for hood time?
The bottom line is that it is up to you. It would be very beneficial to get as much simulated IFR time as possible. It does not matter if it is with your school CFII, a school student, the club CFI, a club student, or an unaffiliated and current pilot friend. Your CFII should not have a problem with that. Instruction (and the cost of it) is about you, not him. Although, they may have a legitimate concern about the quality of training you are receiving outside of the school. And, be wary of training in aircraft that are too different from one another, even in the same make and model.
The benefits of doing IFR time with your school CFII are:

It counts towards dual instruction time.
Your instruction stays consistent in structure.
The CFII will be able to give you the gouge (predisposition of questioning and testing) of the DPE.
The CFII is already familiar with your deficiencies.

The benefits of doing IFR time with the club CFI:

It counts as PIC and IFR time even if it does not count as dual instruction.
Costs less than the school.
You get a different perspective, tools, tips and tricks from an instructor qualified, instrument rated pilot.
A new/different set of eyes may uncover different deficiencies.
Typically, club instructors are going to stick around and have more longevity than school instructors. Although, this may not always be the case. It will give you an opportunity to build camaraderie and community for the future. After all, who is going to give you your Flight Reviews and Instrument Proficiency Checks.

The benefits of doing IFR time with another student:

Another student may have picked up something different but useful in training than you did.
Collaborative learning.
Building community and networking for the future.
Cuts out the cost of an instructor.
If it is a school student, their training will be consistent with yours (supposedly).
If it is a club student, they may have been trained completely differently. You may be able to glean useful tidbits from that different training.
You get to double your total flight/PIC time when trading off safety pilot-flying pilot responsibilities.
Flying! ‘Nough said.

The benefits of doing IFR time with a current and proficient, unaffiliated pilot friend:

Again, Flying! Flying is winning.
Totally different perspective. Remember, a good pilot is always learning, even from other pilots.
Camaraderie, community, networking, friendships, and that $100 burger/breakfast/BBQ.
You can fly any aircraft that you want. Although, it may not count as total and/or PIC time if you are not certificated for that category, class, and type rating if applicable unless the friend is also a CFI.
And of course, money. You can fly something cheap. Ty to find someone who owns their own aircraft.

Lastly, never overlook or underestimate the value of just sitting in the backseat with your headphones on and your mouth shut during another student’s lesson. You may pick up a lot of valuable information when you don’t have the added burden of flying the aircraft. Even if you can’t actually log the time.
As far as the “legal bind” that you mentioned. Any and all pilots acting as PIC will be in a legal bind if something goes wrong. After all, they are PIC for the purposes of safety of flight. That is the responsibility of their certificate. Any PIC who currently or in the future earns a living from their commercial license of any type will have more at stake than a recreational pilot.