You are, of course free to name/call your child whatever you want, but to answer your question "Will Japanese people Kotei is a strange name?", they probably will, but with a big IF - they'll think it's strange if they realize your son has a Japanese name.
Since you're asking the question here, I'm assuming that neither you nor the baby's father are Japanese, nor living in Japan, and so if/when Japanese people meet your son, they won't initially think he has a Japanese name. In other words, they won't think "Oh, his name is the Japanese word Kotei, which means emperor. How strange," so much as "Oh, his name is Kotei. That must be a foreign name," and thus not connect it to Kotaro.
People in Japan would be more likely to think Kotei is a strange name if your son is/appears to be Japanese, since then they probably would connect it linguistically to the Japanese vocabulary word kotei. But while kotei does mean emperor, when people in Japan talk about the emperor of Japan, they usually don't use just the word "kotei." He's more commonly referred to as "kotei heika," "his majesty the emperor," so it's not like hearing the nickname Kotei would immediately have people thinking of the emperor of Japan. They might think of kotei pengin/emperor penguins instead. There's also the chance that if your pronunciation of "Kotei" doesn't have enough length on the o sound, people might hear it as the Japanese word for "fixed" or "set" (固定/こてい instead of 皇帝/こうてい).
On the other hand, if Japanese people do know your son's full name is Kotaro, and you call him Kotei as a nickname, they'll probably think the way you'e shortening the name is unusual. In Japan, the more common ways to shorten Kotaro would be to call him Ko, Ko-kun, or Ko-chan. For a native Japanese speaker, changing Kotaro to Kotei instead of Ko would probably feel a little like how a native English speaker would feel about changing Susan to Susa instead of Susie - sort of strange, but not shockingly so.
「We like kotei spelling better than kotay but if it’s offensive or strange to Japanese people we can stick with kotay. 」
Neither one is offensive. It's not like Japanese people will find it rude and presumptuous that your son's nickname sounds like the word for "emperor." Both will seem strange to Japanese people only if they know the entire backstory of your son's name, but unless you're introducing him as "This is my son Kotaro, but we decided to give him the nickname Kotei" most Japanese people will just think "Kotei must be a normal name in this person's home country."