Normally, when the output doesn't go to a terminal, procps' ps does not truncate fieldscommand lines, but that's unless the $COLUMNS environment variable is set and the procps implementation of watch does happen to set it based on the width of the terminal it is running in¹.
But here, rather than running ps | grep (here using non-standard ps aux API that is actually ignored by busybox ps), you could just do:
watch 'pgrep -af languagetool' (or watch -x pgrep -af languagetool with procps' watch to avoid the unnecessary shell invocations).
pgrep, with -f will match the regexp against the full² process argument list³ only and not report itself or its ancestry (such as the watch command here).
Note that while busybox does have a pgrep applet, it's not always enabled.
Here, to match on command line only without including the sh, grep/awk and watch commands portably between procps and busybox implementations of ps, you could do:
watch ' COLUMNS=0 ps -o args= | awk "/languagetool/ && ! /exclude-me-please/" ' ¹ Shells such as ksh, zsh, bash or fish also set a $COLUMNS shell variable (not when non-interactive in the case of bash) but do not export it to the environment by default.
² Or as much it is possible to obtain as earlier versions of the Linux kernel only made the first 4096 bytes available.
³ Joined with spaces and with transformation of some characters such as control ones similarly to what ps -wwo args does.