If you feel comfortable with opening Macs, I would just give it a try. I have opened up the Mid 2011 iMac and replaced the drive in less than 10 minutes. While it would probably be safer to remove the LCD completely, I have swapped drives without disconnecting any of the connectors on the LCD, I just only lifted it a few inches and used a very short driver.
I am unsure if the fans will be affected by removing the HDD, but like I said, you could quickly open your iMac, but just disconnect the SATA cable plugs, leaving the HDD in, close everything up, and test it see if the fans get all messed up.
As for the fan, I have read that people with the Mid 2011 iMacs have had issues with the fans running full blast with the HDD swapped or removed, and some say that they did not have this issue.
I no longer have my Mid 2011 iMac, so I cannot test it, but either way, I suspect that it might depend on the specific iMac.
Maybe since yours came with a BTO SSD, you will not have the fan issue.
But, I would just try it. You can use iFixit's guide, but just stop at the part where you start disconnecting the LCD connectors, disconnect the HDD, put it back together, and see what happens.
I can replace the hard drive with an SSD
If you iMac is subjected to the max fan speed issue, just swapping the HDD for a SSD or another HDD will not correct the issue. Apple's OEM HDDs have built-in temp sensors, and most third-party HDDs or SSD do not have built-in sensors.
Usual solutions are using an external sensor or using SW to control the fan speed, but I have seen people just short/close the two wires leading to the HDD that are used for the temp sensor.
Swapping with a SSD or just removing the HDD and not replacing with anything would leave the two sensor wires open, which then leads the OS to run all the fans at full blast due to thinking the sensor went bad. This might not happen to your specific iMac, but you can easily check it.
I would just disconnect the HDD to see what happens.