Is there a way to get an Irish PPS Number while living in the UK?
No. At citizensinformation.ie you can learn that "you must be living in Ireland to apply for a PPS number."
You also ask
Can I use my current British EU Health Card in Ireland?
Of course. The UK government has a page about this, which says in part
The European Health Insurance Card (EHIC) allows a British National to access necessary treatment while you are on a temporary stay in Ireland, if you become ill or have an accident. This card should be obtained in advance of your travel.
The UK government actually misstates the situation slightly, because the card is not restricted to British nationals or even EU/EEA/Swiss nationals; it is available to everyone who participates in the NHS. From the NHS page, which is more specific:
Entitlement to an EHIC is based on insurability under EU law, and not on a person's nationality. This applies to all EEA countries. The UK operates a residency-based healthcare system, which means that insurability is generally determined by residency and not by the past or present payment of National Insurance contributions or UK taxes.
Similarly, the mandate of the Irish Health Service Executive is to serve everyone living in Ireland, which you are not.
since the UK is leaving the EU, I’m not sure how much longer this will be valid for.
The rights of EU citizens and British citizens after the UK leaves the EU are a matter of contentious debate at the moment, but the debate largely concerns rights of residence. Other practicalities such as reciprocity of health insurance seem likely to get lost in the mix, which suggests that it's likely that the UK will simply withdraw from the EHIC system either at the moment it leaves the EU or possibly at some later point if there is a transition period. At that point, your most likely avenue for coverage when you are visiting Ireland will be travel insurance.