To have TBE vaccine during allergic times with mild asthmatic symptoms?
Interesting question! 

The Tick Borne encephalitis vaccine is an inactivated vaccine. It is therefore impossible to get the infection from the vaccine. Reports of fever have been noted but that is due to the body mounting an immune response not due to infection. There are therefore no reports of encephalitis or other markers of infection following the vaccine. 

Not only may you have the TBE vaccine if you are asthmatic, but if you are at risk due to your lifestyle and location then, particularly if you have chronic asthma, the TBE vaccine might be a good idea. 

Asthma is usually treated by inhalers and montelukast is reserved for uncontrolled/severe asthma. However in the rarer case of aspirin induced asthma, it is much more first line and used for mild asthma. Montelukast is not considered a mild medication. 

I hope this answers your question :) if not, comment below!