The Sport is a relatively rare vehicle ...
- what are the chances of your finding diff mounting brackets for it?
- what are the chances that those brackets will be attached to a complete differential assembly?
- what are the chances that the complete differential assembly will be attached to a complete vehicle?
If you search the forum(s) for discussions on converting 2WD Sidekicks to 4WD, you'll find that the general consensus is that unless you have a complete donor vehicle to get the parts from, it'll be easier to sell the 2WD and buy a 4WD - that advice doesn't change when the vehicle in question is a 1.8 Sport - sure it'll be harder to find a 4WD Sport to buy, but, it'll be even harder to find the parts that you can't rob from a 1.6 Sidekick, and those parts include the diff mounting brackets.
A quick part number search reveals that the same part number front differential housing was used on ALL the Sidekicks from 89~98, including the Sport models, so yes, you can use the "non-Sport" diff with Sport diff brackets - if you can find them - please note - all I'm discussing is the housing, the internals may be different.