My significant other is a member of a research project that I coordinate. How to allocate funding?
I served for a couple of years on the Conflict of Interest Committee of my institution. During that time I learned many useful and sometimes counterintuitive things about the subject. The principles that should guide you in thinking about what to do to avoid exposing yourself to any accusations of policy or ethics violations are:
You should avoid putting yourself in a conflict of interest with respect to the grant and your significant other, if at all possible.
The point of the “if” qualifier is that, as you seem to believe, it may not in fact be possible to avoid without taking unreasonably drastic measures such as you or your SO resigning your roles.
You should avoid creating the appearance of an unmanaged conflict of interest, period. No exceptions.
The point of “appearance” is that it’s not just the conflict that needs to be avoided but also the appearance of one. You may be the most ethical person on the planet, or you may even know certain facts that not everyone knows that explain away the conflict and make you confident what you are doing is fine, but the appearance of a conflict can invite consequences that are just as bad, and possibly worse (e.g. a headline in the New York Times followed by calls for your resignation, a public lynching or whatnot), than the conflict itself.
The point of “unmanaged” is that if you can’t avoid a conflict - and one of the surprising things I learned is that conflicts can not always be avoided, nor is avoiding all conflicts at any cost even a desirable goal to have - then the next best thing is to find a way to manage the conflict. For example, at many institutions one can set up a conflict management plan, which may involve additional levels of oversight (for example a department chair or dean signing off on your funding allocation decisions pertaining to your significant other, and maybe additional reporting requirements such as you filing a letter of explanation justifying any such decision).
This goes without saying, but you should find out all applicable rules and policies that govern your situation.
The bottom line is, you should ask for guidance from your institution and your funding agency on how to address the situation. Have in writing documents, written (or at least reviewed and approved) by people in authority, which specify how you are expected to handle the situation. Then follow what these documents say.
However, I'm now in a relationship with one of its members. How should I proceed regarding allocating funds to my significant other?
The best thing to do here is to disclose the conflict-of-interest to your Head of School** and ask that they allocate a different academic to decide the funding allocation. The Head of School might have a different idea, and might decide that they want you to make the decision anyway, and if so, they will direct you to do that. By suggesting you be removed from making the decision you will have given yourself protection in the event that you are asked to make the decision and then later accused of bias. That will put you in a strong position if you are later accused of any mismanagement or bias.
I would like to declare a conflict of interest, but there does not seem to exist a way to deal with this conflict without some drastic consequences.
That is ridiculous. There certainly should not be any adverse consequences merely for declaring a conflict-of-interest prior to making a decision. The whole point of a conflict-of-interest policy is to encourage people to disclose potential conflicts-of-interest so that alternative decision-making arrangements can be made to avoid the conflict. The onus is on you to declare the conflict to your Head of School, and the onus is then on the Head of School to decide how to proceed. If your university has a conflict-of-interest policy then you should read it and follow its reporting rules, and if there is no such policy then you should declare the conflict directly to your Head of School with a simple email.
Also, you are not required to find a solution to your own conflict-of-interest. The whole point of the conflict is to make your decision-making suspect, and thereby taint decisions you make on the matter. Once you have disclosed the conflict-of-interest to your Head of School, the onus of decision-making will be on him/her to decide if and how they would like to alter the normal decision-making mechanisms.
** Throughout this answer I will refer to disclosure to your Head of School. Depending on your level in the university it might be sufficient to make disclosure to your own supervisor, even if this staff member is lower than Head of School. For simplicity, I will refer to disclosures to HoS.
You report the conflict of interest to your superior, and ask for help developing a management plan moving forward. Full stop.