There is no easy solution. You could turn on clobber mode and COPY will prompt you before overwriting an existing file.
Otherwise you'll have to write a custom COPY script that checks for an existing file and formats a new name based on the existing name.
This isn't checked, but should be close. You'll probably need to do something with quotes too if there are any spaces.
iff ISFILE %dest then
do i = 1 to 65535
if ISFILE %@PATH[%2]%@NAME[%2](%i).%EXT[%2] ITERATE
copy %1 %dest