For now eight does not allow outgoing SMS to foreign numbers.
Starhub is the only one to offer such SMSs for free, on their 5G Unlimited+ Plans ($38 & above).
On the rest it costs around $0.10 - $0.20 per SMS. Singtel has a Global SMS add on pack for $3 per month that allows you to send 50 SMSs.
Your choice is to sign up for a postpaid plan with the 3 MNOs (Singtel, Starhub, M1), MyRepublic or the Singtel MVNOs that support global SMS.
For your use case, SIMBA may be most appropriate, as they offer a $10 plan, but SMSs cannot be sent while roaming, they can only be sent while in SG.
If you require sending international SMSs while roaming, you will need to subscribe to the remaining options.