Use this perl one-liner:

echo ">BNY.1.2.t17987.mrna1 CDS=1-1065\nACTG" | \
    perl -lpe '
s{\s.*}{};
s{[.][^.]+\z}{};
'

In the example based on the header you showed, it prints:

>BNY.1.2.t17987
ACTG

Here, these command line flags are used:

-e: tells the Perl interpreter to look for the code in-line instead of in a file;
-p: loop over each line of the input file or, if none, STDIN; assign the line to $_, execute the code provided, then print $_;
-l: (lowercase "L"): strip the input line separator (default "\n" on *NIX), then add it during printing.

The code does 2 substitutions, each by default done on $_. The first one replaces everything after the first whitespace with an empty string, which trims " CDS=1-1065". The second one trims everything after and including the last "." until the end of the line ("\z").