Will frequent small select queries slow down frequent write queries in InnoDB table in MySQL?
The only way to know is to benchmark it.  If your query is fast, the chance of any notable impact on other queries is low, assuming your database server has sufficient disk bandwidth, multiple CPU cores, sufficient memory, and an appropriate innodb_buffer_pool_size setting for both the working dataset and the hardware.

If you don't already have a replica ("slave") server, then you should consider implementing that -- a slave server imposes a relatively minimal load on the master, the select queries could be run against the replica, and, of course, you would then have an alternate server you could use in the event of a catastrophic failure of the master server.

Use EXPLAIN SELECT ... to understand how the server is actually processing your query, and then consider whether it needs to be optimized or rewritten, particularly if it is doing things like full table scans or temporary tables.

When benchmarking your query, remember to use SELECT SQL_NO_CACHE ..., otherwise the performance may be deceptively fast if none of the underlying tables has changed between subsequent invocations.