I'd be genuinely interested in understanding how one can reconcile conservatism with compassion and empathy

Let's take an example.

"A slightly different case is that of businessman John Timpson, who is the controlling shareholder in an eponymous chain of shoe repair and key-cutting stores that dot shopping malls across England. Over the past six years Timpson has donated 478,000 pounds to his son, Edward, a Conservative member of parliament, via his company, Timpson Ltd, records show" (reuters)

On the other hand they have always had a policy of accepting ex-offenders as employees, which you might think a bit 'bleeding heart' leftie thinking.
"We pay our colleagues as much as we can afford rather than as little as we can get away with"
"Timpson really are an equal opportunities employer. We consider anyone for our vacancies as long as they are able to do the job. This includes ex-offenders and other marginalised groups."

These are not just empty words: it can be backed up by seeing how many ex-offenders they have compared to other businesses.