Figured it out finally so I'll answer myself, setting a custom color scale like below then assigning it to the argument would work:

colorscale = [
[0, 'rgb(38, 53, 113)'], 
[0.5, 'rgb(57, 162, 225)'],
[1, 'rgb(234, 32, 41)']
]

...and then in the argument:color_continuous_scale = colorscale

It seems that your best option for showing every color is:

fig.data[0].contours.coloring = 'heatmap'

Code:

import plotly.graph_objects as go
fig = go.Figure(data =
    go.Contour(
        z=[[10, 10.625, 12.5, 15.625, 20],
           [5.625, 6.25, 8.125, 11.25, 15.625],
           [2.5, 3.125, 5., 8.125, 12.5],
           [0.625, 1.25, 3.125, 6.25, 10.625],
           [0, 0.625, 2.5, 5.625, 10]]
    ))

fig.data[0].contours.coloring = 'heatmap'

fig.show()

The code you use to generate a color palette works just fine. You only need to provide data that match with a heatmap. The following code provides this:

library(RColorBrewer)
library(plotly)

# your palette definition
palette <- colorRampPalette(c("darkblue", "blue", "lightblue1",
                              "green","yellow", "red", "darkred"))

set.seed(9876)    # for reproducibility

## a complete random set
hmdata <- matrix(data = sample(x = 1:10^6, size = 100*100), nrow = 100, ncol = 100)
plot_ly(z = hmdata, colors = palette(50), type = "heatmap")

This gives the following heatmap:

## a random set that has been sorted
hmdata_s <- matrix(data = sort(sample(x = 1:10^6, size = 100*100)), nrow = 100, ncol = 100)
plot_ly(z = hmdata_s, colors = palette(50), type = "heatmap")

Yielding this plot:

Please let me know whether this is what you want.

UPDATE

you can set custom scale in plot_ly with zauto, zmax, and zmin. The following 2 pieces of code and graphs will illustrate this:

The scale is set from 1 to 100 and the data vary similarly:

hmdata_s3 <- matrix(data = sort(sample(x = 1:100, size = 100*100, replace = TRUE)), nrow = 100, ncol = 100)
plot_ly(z = hmdata_s3, colors = palette(50), type = "heatmap", zauto = FALSE, zmin = 1, zmax = 100)

The scale is set from 1 to 100 and the data vary between 50 and 100 only

hmdata_s4 <- matrix(data = sort(sample(x = 50:100, size = 100*100, replace = TRUE)), nrow = 100, ncol = 100)
plot_ly(z = hmdata_s4, colors = palette(50), type = "heatmap", zauto = FALSE, zmin = 1, zmax = 100)

trace0 = go.Scatter(
    x = foo,
    y = bar,
    name = 'baz',
    line = dict(
        color = ('rgb(6, 12, 24)'),
        width = 4)
)

This allows you to change the color of the line or you could use

colors = `['rgb(67,67,67)', 'rgb(115,115,115)', 'rgb(49,130,189)', 'rgb(189,189,189)']`

for separate lines of a graph. To use the specified color gradient try

data = [
    go.Scatter(
        y=[1, 1, 1, 1, 1],
        marker=dict(
            size=12,
            cmax=4,
            cmin=0,
            color=[0, 1, 2, 3, 4],
            colorbar=dict(
                title='Colorbar'
            ),
            colorscale='Viridis'
        ),
        mode='markers')
]