There's not much to it. The downsize option will affect image eventual size. Bilinear would be a good option, but if you keep the original and work on a copy, you can always go back.
Having downsized the image, Save it as a JPG with a quality setting of 10, and see what the resulting image size is. If it exceeds 300kb the Save as JPG again, and use a lower quality setting.
Note that JPG compression works better on images with fewer variations on colour. A solid red will compress to a tiny size, where as a psychedelic mess will save as a much larger file.
[EDIT] Note that my screen shot has Bicubic Sharper highlighted. Bt all means try that, but it will result in a larger file size than Bilinear. I also find that Bicubic Sharper is much too harsh and produces an oversharpened end result.
I have a large image 9,197KB (4928px x 3264 px).
I want to crop to 1600 x 200 150 dpi with a file size between 200-300 kb in jpeg.
Lets examine what you have and want to do.
"I have a large image 9,197KB" That is more or less meaningless for you most likely referring do a file size of some image file format which is most likely data compressed.
"(4928px x 3264 px)" Now that is the Image size you have which has an aspect ratio of 1.509803921568627 very close to a standard 3:2 Aspect Ratio image ratio of of 1.5.
You want an image 1600x200px which has aspect tario of 8:1 which is very wide compared a standard a 3:2 aspect ratio 1.5. Such a crop would drastically change your image's composition because of the aspect ratio crop. You would be discarding the majority of your Image's content.
You would be discarding the white areas and keeping the white area on an images depicted here. It would not be very satisfactory you would be discarding most of you images content. Are you sure this is what you want to do. It would be easy to do but IMO not something you would want ti do.
Yes, that is what I want to do - it is for a website banner.
By George I think she's got it! Thanks
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One additional comment from me.
JJMack has covered cropping very nicely into the ratio you require (8:1). Do this with "Delete Cropped pixels" unchecked in the options bar and save as a PSD. That way you can go back later and adjust the crop differently if you require.
Now go to File >Export>Save for web legacy. Select jpeg and put in your required pixel dimensions. Then adjust the quality slider to meet the file size that you require. The file size is shown under the image preview.
In short - crop non destructively - save the master as a PSD which means you retain the layers and the ability to uncrop or change the crop later. Then Export a copy as a jpeg in the pixel dimensions you need.