I wouldn’t worry about “making a fool of yourself” either. You cannot be anything other than yourself.
For me, it’s the hymns which set me off or observing grief on the part of others.
Last December at Paws Requiem Mass I reckon I held things together quite well, just let the tears flow silently and honestly was more supported by the presence of our friends than upset by their grief. TBH I don’t notice except for the very special friend who did the eulogy and who, bless him, was on the verge of tears throughout I think because he had our DGS aged 7 right in front of him in the front pew with his Mummy and Daddy and me, poor little boy and was in tears, , but did it so silently I didn’t notice.
A good funeral (?) should also be a celebration of a life with anecdotes to raise a smile or the memories of the good times.
Don’t worry, weep, smile, remember and (what I did) a nip of medicinal brandy should not render you unfit to drive.
Weep if you must
Parting is hell
But life goes on
So sing as well