What rights do I have to enforce the payment method of the settlement agreement?
Although you use the past tense in "Previously the wording [...] included", I gather that the record that memorializes the settlement has not been modified.
It is important to ascertain who proposed or drafted the wording "paid to Party 1 or to us as their solicitors" (the OR clause"), since that would determine whether you may resort to the doctrine of contra proferentem and therefore be entitled to decide to whom you opt to make the payment. Items 12.65-12.67 here reflect that contra proferentem is cognizable under Singapore law.
But note that item 12.66 highlights one complication of which you need to be aware (and to overcome):
if the contract were bilaterally negotiated, then the contra
proferentem rule would not apply [...] where the clause 'emerged as a
result of joint efforts' [...] [or] a joint drafting effort.
This is a clear-cut example of absurdity (on the part of whoever came up with those holdings) because many, many contracts are the result of negotiations --thus amounting to joint efforts--, and yet that fact does not imply that the parties to such contracts have comparable bargaining power. And not only a settlement is a typical example of "bilaterally negotiated contract", but the process and/or confidentiality of settlement fine-tuning makes it harder to prove who proposed what.
That being said, two details suggest that Party1/its solicitors (not you) proposed/drafted the OR clause: (1) the language "to us" in reference to one of the alternatives about payment, and (2) the rarity for a payer (unless his bargaining power is significantly greater) to impose a list of options of to whom exactly he may choose pay. If Party1 refuted in court the application of contra proferentem, you might want to argue your position from this angle.
Another relevant but unclear detail is whether or not the settlement truly contains any "instructions" pursuant to payment to an offshore-account. If it does not, then Party 1 would have a harder time arguing the legitimacy of his post-settlement preference.