a few thoughts from an obs anaesthetist PoV...
codeine isn't used post partum in the UK for women who are planning to breast feed. Dihydrocodeine is the medium-strength opioid more routinely used because it's safe to use whilst breastfeeding. The two drugs are different despite having similar names and some people get on better with one vs the other.
There will be diamorphine in the spinal mix - this helps to ensure that the spinal is dense enough and works long enough to do the operation. It also lasts longer than the local anaesthetic element of the spinal and helps with early post op pain relief. Spinal diamorphine doesn't cause any of the systemic side effects like drowsiness etc.
A diclofenac suppository is fairly routinely given in theatre unless there is a contraindication. For the ongoing post partum period there isn't much difference in terms of analgesia between diclofenac and ibuprofen for most people, but diclofenac is more likely to cause side effects.
TAP blocks mentioned by PP may be an option in theatre, but not normally as an ongoing thing. Not all of us do them routinely though so that would be dependent on who's doing the list.
PPs idea of suggesting asking for the spinal to be left running suggests a complete lack of insight to what a spinal is (spinal catheters are rarely used in UK practice - it's a single shot injection).
If for any reason a spinal (or less likely and epidural) isn't possible then a GA would involve needing opioid medications intraoperatively as the pain relief is much trickier at least initially than after a spinal c-section.
There is surprising variability in women's pain relief requirements after a c-section - some really are ok with paracetamol & ibuprofen but many women need stronger stuff. To some extent in may depend on the reason for the section, but it's not that predictable. It is worth remembering that uncontrolled pain is physiologically not good for you, so if pain is bad it's usually far better to take appropriate analgesia (+/- anti-sickness medication if that's a problem) than to be immobile in bed, not breathing deeply, brewing a DVT, chest infection, UTI etc.
Definitely talk to the anaesthetist - there may be things they can do differently/tweak, but try to keep a slightly open mind. By declining any opioids you may be committing yourself to a pretty rubbish experience, and all for something that may not actually be a high risk problem (as people with advanced cancer are very different to the demographic of people having c-sections, the side effects/risks vs benefits can be quite different too). HTH.