How do I get a passport stamp when traveling from Ireland to the UK?

How do I get my passport stamped with an entry stamp to UK from Republic of Ireland?

You don't. If you try, they will refuse because leave-to-enter can only be issued at an authorised primary control point and those places cannot be approached from inland. Moreover, it's a sackable offence to 'stamp someone in' inland (that can only be done by applying to the Home Office).

If there are any date calculations to be done they will use the start date on your ROI entry stamp. If you have not been in the UK previously, and you are in the UK more than 90 days beyond the ROI entry stamp, you will become either an overstayer or illegal entrant (depending upon how they caught you).

I need to be able to show my exit from Ireland. I'm not an EU/EEA citizen.

Why? If your leave in the ROI has been entirely spent and you go to the UK (as pointed out above), you are automatically an overstayer or illegal entrant. The UK Border Force uses the original entry stamp into the Common Travel Area as their reference point. On the other hand, if you have unspent leave there is no need to document anything.

The canonical advice if you absolutely need to document your arrival in the UK (i.e., a spouse or student or work permit holder) is to channel hop outside the CTA, France or Belgium being the best options.

NOTE: You can see the chart from their operations manual at: How long before I can re-enter the UK after Youth Mobility? the red thingies do not apply to your case. Based on what you wrote, your path ends at 'deemed to 3 months Code 3'. The 3 months is from the date you entered the CTA (you do not 'automatically' get an extension).

NOTE: For completeness sake... Yes, as mentioned above you can apply to the Home Office. Doing this route is wildly adventurous and emphatically not recommended without a solicitor making representations for you.

NOTE: this answer is time-limited and heavily affected by Brexit. ILPA prepared evidence for the House of Lords’ Select Committee on the European Union in a briefing called ILPA Evidence for the House of Lords’ Select Committee on the European Union for its enquiry into the impact on the relationship between the United Kingdom and Ireland following the vote by UK citizens to leave the European Union for a list of impacts.

See also Agreement for the sharing of visa, immigration, and nationality information