The solution I ended up with was not at all what I had originally envisioned. While I do use the ContentService to get the content, the actual updates are done using more generic methods and serializing/deserializing Json. For what it's worth, here is what is working for me to get BlockList content, do some simple find and replace, then update and save the BlockList content. I've added a way to handle both single content and nested content (eg, Lists of BlockList content). I'm sure that can be refactored to be cleaner.

I am more than open to a better way to do this!

private readonly IContentService _contentService;
...
public FindAndReplaceApiController(IContentService contentService)
{
    _contentService = contentService;
}
...
// your content id
var content = _contentService.GetById(0000);  
// the property alias for the BlockList content from the Document Type
var rawJsonBlock = content.GetValue("content").ToString(); 
// the entire BlockList Json block
BlockValue blockValue = JsonConvert.DeserializeObject<BlockValue>(rawJsonBlock);
// Note: using System.Text.Json throws exceptions when deserializing OR returns an empty object, I don't know why

foreach (var column in blockValue.ContentData)
{
    foreach (var columnProperty in column.RawPropertyValues)
    {
        // the Value is a text string, unless it's a BlockList, in which case it's a Json object
        var block = columnProperty.Value; 

        if (block == null) continue;

        // check if the Value is a BlockList, for repeating BlockList items
        if (block.ToString().Contains("Umbraco.BlockList", StringComparison.InvariantCultureIgnoreCase))
        {
            // if it is, we can deserialize it to a BlockValue since we have nested content
            var innerBlock = JsonConvert.DeserializeObject<BlockValue>(columnProperty.Value.ToString());

            foreach (var innerBlockValue in innerBlock.ContentData)
            {
                foreach(var blockProperty in innerBlockValue.RawPropertyValues)
                {
                    Debug.WriteLine($"Inner Block Value: {blockProperty.Value}");

                    if (blockProperty.Value.ToString().Contains("some old value", StringComparison.InvariantCultureIgnoreCase))
                    {
                        // replace the old with the new!
                        innerBlockValue.RawPropertyValues[blockProperty.Key] = blockProperty.Value.ToString()
                                                                                .Replace("some old value"
                                                                                , "some new value"
                                                                                , StringComparison.InvariantCultureIgnoreCase);
                        // serialize the updated value to Json for storage
                        column.RawPropertyValues[columnProperty.Key] = JsonConvert.SerializeObject(innerBlock);
                        content.Properties["content"].SetValue(JsonConvert.SerializeObject(blockValue));
                    }
                }
            }
        }
        else
        {
            // in this case, the block content is not nested
            Debug.WriteLine($"Top Level Block Value: {block}");

            if(block.ToString().Contains("some old value", StringComparison.InvariantCultureIgnoreCase))
            {
                // update the block object directly
                block = block.ToString().Replace("some old value", "some new value", StringComparison.InvariantCultureIgnoreCase);
                // serialize the updated value to Json for storage
                column.RawPropertyValues[columnProperty.Key] = block;
                content.Properties["content"].SetValue(JsonConvert.SerializeObject(blockValue));
            }
        }
    }
}

if (content.IsDirty())
{
    // save the updates, if any
    _contentService.SaveAndPublish(content);
}
...