Can bacteria resistance become a problem from stopping doxycycline hyclate after only 6 of 20 doses due to a mild/moderate allergic reaction?

It can: Incomplete antibiotic courses are known to cause resistance of bacteria to the antibiotic. Doxycycline is a bacteriostatic antibiotic, meaning that it typically suppresses bacteria more so than killing them would probably be more prone to that than other antibiotics. Also, it depends on the bacterial propensity to develop resistance as some do so more than others.