I would say there are 3 main difficulties:
AT power connector, versus ATX power supplies. As you've noted, adapters are available to convert from ATX to AT, you will just need to figure out what to use as a power switch. The AT used the main power supply disconnect as the system power switch, while ATX moved to 'soft power' via a motherboard connector.
Some issues you may encounter are:
Motherboard size. An AT motherboard is 13 inches from front to back,
while the ATX standard mandates 9.6 inches. If a case takes an
Extended ATX motherboard it should take an AT one.
Mounting pillars / holes. On an AT motherboard the position of the holes is not precisely defined by the standard, whereas ATX is more ...
For a while, many PC cases were made to be both AT and ATX compatible. This would have been in the mid-1990s, during the changeover from Socket 7 to Slot 1 (Pentium II) and Super Socket 7 (AMD K6 series) when motherboards were made in both shapes. You could try looking for cases from that period, or from the early 2000s.
AT motherboards (in practice, ...