How well does my model fit? Specifying a null-model in non-linear mixed models

Question

I want to fit a model y ~ b * exp(-exp(a) * x), but including a random effect, with this data:

# data:
data = structure(list(re = structure(c(1L, 1L, 2L, 2L, 3L, 3L, 3L, 2L, 
2L, 3L, 1L, 1L, 3L, 3L, 2L, 2L, 1L, 1L, 2L, 2L, 3L, 3L, 1L, 1L, 
2L, 1L, 1L, 3L, 3L, 2L), levels = c("A", "B", "C"), class = "factor"), 
    x = c(0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 
    3, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8), y = c(0.0899425978552549, 
    -1.85384333820306, -4.14514965495534, -1.81807449635766, 
    -1.35326955027327, -1.28825429472141, -0.209944981000945, 
    -2.55036342588007, -2.41329880502537, 1.60816564193517, 0.974291267956421, 
    -1.03569994133107, 0.259011545358077, 1.15202768590689, -1.11099110226214, 
    0.723503984838873, 1.0498337589214, 0.0331911544521152, -1.0534870983813, 
    -2.39106769323274, 0.145263217335939, -0.314587019845322, 
    1.27885667618699, -1.46171756397001, 0.69031827310659, 0.656834906437993, 
    0.0910321826159956, 1.59640958118439, 1.02260280171595, -0.0386924537729473
    )), row.names = c(NA, -30L), class = c("tbl_df", "tbl", "data.frame"
))

Thus I would fit this model with nlme:

# define the model function
f = function(x, a, b, Asym = 0) Asym + (b - Asym) * exp(-exp(a) * x)
define the model:
model.1 <- nlme(y ~ f(x, a, b), 
               data =  d_pub,
               fixed = a + b ~ 1,
               random = b ~ 1 | re,
               start = c(b = -0.15, a = -0.7))
model:
summary(model.1)
# Nonlinear mixed-effects model fit by maximum likelihood
# Model: y ~ f(x, a, b) 
# Data: d_pub 
# AIC      BIC    logLik
# 102.6748 108.2796 -47.33741
# 
# Random effects:
#   Formula: b ~ 1 | re
# b Residual
# StdDev: 1.10461 1.095258
# 
# Fixed effects:  a + b ~ 1 
# Value Std.Error DF   t-value p-value
# a -0.6980492 0.6247593 26 -1.117309  0.2741
# b -1.4309434 0.7840392 26 -1.825092  0.0795
# Correlation: 
#   a     
# b -0.215
# 
# Standardized Within-Group Residuals:
#   Min         Q1        Med         Q3        Max 
# -2.0549776 -0.5476474  0.1779084  0.9287772  1.8802465 
# 
# Number of Observations: 30
# Number of Groups: 3 

I would then also quickly look at the residuals:

plot(model.1)
qqnorm(resid(model.1)); qqline(resid(model.1))

If I look at the data (and the model prediction and the std error of the model parameters), I don't actually assume that this model is particularly good. In this particular case (although in other cases the model seems to fit quite well), it might even be possible that y doesn't depend on x at all. To test this, I would like to compare it to a "null-model" to see whether it performs better or not.

Intuitively I would have fitted a linear model: lme(y~1, data = data, random = ~1|re), but then I wouldn't know how compare these very different kinds of models!

Update: I have now fitted the null-model as following:

f2= function(x, a) a
model.0 = nlme(diff ~ f2(x, a),
                          data = d_pub,
                          fixed = a ~ 1,
                          random = a ~ 1|re,
                          start = c(a = mean(d_pub$diff)))
anova(model.1, model.0)

Is there any way to assess whether a nlme-model fits the data, some measure of gooness-of-fit? And are there practiacal ways to also test model assumptions for nlme-models?