On a general and administrative basis, reusing the same private key for distinct usages is bad.
For this specific case, this should be fine (although you should use SHA-256 instead of SHA-1, if only for reasons of public relations: SHA-1 has bad press and any use of SHA-1 for a new design will be questioned), subject to the important caveat below.
The current date is a monotonous scale. However, your device does not have direct access to that physical value; it only has a clock which provides a discrete (it works by small increments, e.g. 1 second each) approximation of the current date. If you try to generate your AES key twice in the same second, and the "current date" changes only once per second, then you get the same seed, hence the same output, which can be bad. Also, more importantly, if the user (or attacker) can set the clock, then he can potentially enforce a seed reuse, leading to a PRNG sequence repeat. "Setting the clock" must be thought in a wide sense: an attacker who have the device in his hands may try to freeze or cook it, because quartz are sensitive to temperature. He may also play with the power input, e.g. adding a high frequency signal into it, on the hope that it would interact with the quartz and change its oscillating frequency. Possibilities are quite endless; the bottom line is that it is very hard for an embedded device to have an accurate, secure notion of the current time. Even getting a value which an attacker cannot force to repeat can be quite a challenge.
A more robust version would use a counter, which you increase every time you need a new AES key. Make sure that the counter is large enough that it cannot overflow (e.g. use a 64-bit counter). The counter must have permanent storage and never be reset throughout the lifetime of the device; you must also update the counter storage before using the counter value in the PRNG.
By the way, using the RSA private key as PRNG seed will not help you when it comes to actually generating the RSA private key itself. So either you generate the RSA private key externally and import it into the device (e.g. as part of factory initialization), or you have a problem. This might lead to another problem, depending on your exact situation: if a user can force the value of the RSA private key, then he can predict the PRNG output. You do not give information on your security model to know whether this can be an issue for you.