How Do I Design a Search Service That Handles Traffic and Is Cost-effective

Hello everyone, I am a junior developer at a small ecommerce company. I've been tasked with designing a search service for their shopify Site and I am trying to design how to make it so it can handle at least 300 concurrent users and is also fast. The search logic itself will reverse-index the products and later i'm told there is a semantic layer that perhaps needs to be included.

There aren't really any senior engineers in the company so I have to figure this out myself and I don't have any experience with system design.

Since it is shopify app I am using React for the UI for the search feature and Express for the backend which will first grab all the products from shopify(around 10k) and either store it in server memory or a database(MongoDB). This full stack app is embeded within their storefront on Shopify. I have been going over design iterations with claude and Antigravity but both of them recommended me to put the products in memory and, if I make the search logic fast enough and reverse-index the products, the server will be able to handle it. I kept arguing because I always read that applications should be asynchronous by default but it kept arguing back that the overhead of creating async promises would be far larger cost than letting the CPU handle the searching, and how even though it is blocking operations it will be done in a tenth of a milisecond. I am not very trusting of this, and I don't want to commit to such an action, so I am wondering how I can go about this or what I can look into. I was instructed to keep the cost minimal so I assume less than 50$ a month altogether. I am also unsure of normal costs and how much a good search feature would cost on average.

Requirements:

-search should handle high load and be decently quick.

-search backend must first pull from a shopify API and then sync the products

-search must not accumulate a giant cost, less than $50/month(what I was told ) for any services or insfrastructure fees

what are some things I can look into, what is some advice.