I've done a lot of research into this in the past, most recently late last year when I was trying to book an international flight.
Searching on the web, I find conflicting information regarding Secure Flight needing the suffix in your name. I found a number of sources that said to omit it. Some said it was needed. At least one said it initially was not, but now it is. Any links to referenced documents on tsa.gov are now dead, and there is no useful information on the site (as of the last time I checked).
In this case, I booked without it. I had no problems at any point in the travel process. Neither did my friend I was traveling with, who also has a suffix on his passport.
Some data points:
1) I have the suffix in the suffix column on my AA FF profile. I cannot manually add the suffix to my last name when booking a ticket on aa.com; the system tells me it doesn't match the name on my account. When I've needed to book with it, I've had to call AA and have them manually enter it.
2) I have booked both ways in the past, either last name or last name suffix (no space) on the ticket. I haven't had an issue with PreCheck in either case; the only problem I ever encountered was the old US auto-upgrade system would refuse to upgrade me, claiming my name didn't match my FF acct. That was eventually fixed. (My US acct also had the suffix in the suffix column.)
3) My passport has the suffix, and my GE application did as well, but it's not on my GE card.
Annecdotally, I've rarely encountered an IT system that knows how to handle the suffix properly. When people would send me e-mails at my previous employer, the LDAP system would auto-complete my name with a seemingly random combination of first name, last name, and suffix. My personal favorite was when I received an e-mail addressed to "III III III."
Best of luck,
OM