Offer new product to current clients at discount?

If you just want clients for testimonials you can do this. It not a bad idea to get started. However let me propose that you may not need to give a discount. Those who have already spent money with you have high percentage chance of doing it again. I believe you can get full pay and they would be glad to pay it if it is a viable service. You have already built value with these people.

However if you must,maybe give the first month half price for current clients only that sign up by such and such date. Create a since of urgency and a call to action.

If its a product with value they dont need an open invite without a call to action. Real value and the fact they they are getting a deal no one else get needs to have conditions to it. Such as time frame to opt in before the deal closes.

We include reputation management in many of our packages. Ranges from an additional $250 to $1000 per month depending on size of company and what we are reporting.
I agree with starting at lower price since this is new to you and then ramp it up.

Typically I'll occasionally check on my existing customers web presence. If I see something that should/could be corrected I'll just say "Hey, did you know ..." Maybe a typo, address correction or a bad review.

They'll usually ask if I can fix and I get the gig. I charge by the hour with no on going commitments but of course things aren't usually resolved in 1 month so they'll buy another hour or so depending what the issue is. I'm pretty laid back with my clients but I get referrals and they don't hesitate to call me with questions or things they'd like done so this style seems to work well for me.

Best of luck with your rep mgt biz.

Offering a lower introductory price could bring in more business for you. You just may have to alert them that this is for a limited time and eventually the price will rise.

All good point made above. The limited time with you existing clients is key. For you existing clients don't be afraid to call and ask for a sit down. Bring them some coffee and show them what you have been working on. You can even let them know that this is the latest and greatest and you are giving it to a limited amount of you clients (they don't need to know if you have 1 or 100) for a reduced cost. If they ask why, tell them it's because you would like a testimonial after and the possibility for them to speak with a potential client.
Until you get it really rolling make it about them. I have done some pro-bono work for hair salons and car washes. Website that only took me a few hours, gave me great exposure, and saved me money in the end because I was going to use them anyways. From the hair salon I landed a $700 SEO gig from a lawyer that goes in there and started talking about needing his website to get found. Salon person says "O I know someone"

Good luck.