https://clickamericana.com/topics/science-technology/vintage-portable-manual-typewriters
The Smith-Coronas were offered in 4 different colors.
The Remington Quiet-Riter was eventually offered in white sand, desert sage, mist green, and French gray,
The Royal HH was offered in 6 colors including green, pink, and blue. Brown was the most ubiquitous.
https://oldbobsoldtypewriters.com/portfolio-item/1955-royal-quiet-de-luxe-blue/
Royal Quiet De Luxe in ankony Blue (AB)
If we're going to do history, let's get it right. The Royal HH and REs offered color in the '52-54 range, but they were the exception rather than the rule. These are very difficult to find now. The Quiet De Luxes didn't get color until into 1956 and continued until 1958 when they were replaced by the (also) colorful Royal Futuras.
There were pockets of the late 20s and early 30s when Royal and Underwood among others experimented with color on portables as well, but these tended to be more basic reds, blues, and greens. The Corona 4s models were finished in DuPont DUCO® from 1927 to 1939. In a similar time period the Royal P also came in a small variety of colors.
Royal QDLs from '56-59, though advertisement more likely from '57-58 due to the white.
Royal Green, Royal Star White, Royal Beige, Royal Gray, Royal Red, Royal Turquoise,
Add specifies the QDL is available in Canterbury Pica or Canterbury Elite.
https://uppercasepublishing.squarespace.com/typewriter
The Royal FP typewriter came in five different colors including pearl gray, sandstone (yellow), willow green, sea blue, and cameo pink.
In the late 1950s Royal released versions of the Royal Quiet De Luxe in 6 different colors.
If you're looking for pink, here's a shop that repairs and refinishes them (repaints) in pink: https://karlbusinessmachines.com/pink-typewriters/
As for natural pink manufactured typewriters, you're probably primarily looking at the Royal Quiet De Luxe (circa '55-58), Royal Futura 800, Royal FP (a larger heavy, standard desktop machine), the late 50's Smith-Corona 5 series (Clippers, Sterlings, Silent Supers). There were a few later in the 60s and 70s, but they're more difficult to find in the US and aren't as solid. These aren't rare typewriters, but generally because the collector's market likes color, you should expect to pay a $50-150 premium for pink in most online markets. Professional shops are much less likely to charge this sort of premium on pink unless it's for a machine they've had to strip down and repaint. (See also: https://www.reddit.com/r/typewriters/comments/1kw5jxh/what_are_your_thoughts_on_these_and_the_prices/)
For colloquial experience from a typewriter hunter from the Los Angeles area, I'll say I've been looking for a Cameo Pink smooth Royal FP for over a year and finally found one in solid, but restorable shape for $35, but it's taken a year of searching and it'll take a half-day's labor and some parts to clean it up to what a professional shop would consider solid working condition. In that time I've seen maybe half a dozen in online shops sell them in similar rough condition going for anywhere between $100 and $300. For your budget range, getting one from a professional shop is really the best, most economical way to go.
Please note that you positively don't want any of the new pink plastic typewriters made by "Royal Classic" or "We R Memory Keepers" (etc.) which you'll find on Amazon, Michaels, Home Depot, etc. for $200-300. They are dreadful machines, and if you really want one, save the money and buy one from ShopGoodwill.com for $5-15 instead.
If you wanted to go custom in your particular budget range, you might check out Gerren's work at Hot Rod Typewriter Co. where he does some truly gorgeous paint jobs on excellent typewriters. I suspect a pink Olympia SM3 from him would be a thing to behold.
Reply to u/dtja1l at https://old.reddit.com/r/typewriters/comments/1l0l6pi/help_finding_a_thoughtful_typewriter_as_a_gift/ with regard to buying a pink typewriter for a birthday present
https://www.reddit.com/r/typewriters/comments/1kppij9/blue_olivetti_valentine_how_rare_is_it/
In increasing order of rarity, the Olivetti Valentine was made in red, white, green, and blue.